Repunit

Repunit ist ein Kofferwort aus den englischen Wörtern repeated (wiederholt) und unit (Einheit) und bezeichnet eine Zahl, die nur die Ziffer 1 enthält. Eine Repunit ist eine besondere Repdigit („Schnapszahl“); die Bezeichnung Repunit wurde 1966 von Albert H. Beiler geprägt.^[1] Im Deutschen wird auch die Bezeichnung Einserkolonne oder Einserschlange verwendet.

Eine prime Repunit oder Repunit-Primzahl ist eine Repunit, die zugleich eine Primzahl ist.

Definition

Mathematisch sind Repunits (im Dezimalsystem) definiert als

R_{n}={\frac {10^{n}-1}{9}}=\sum _{k=0}^{n-1}10^{k}=\overbrace {11\dotso 1} ^{n{\text{ Ziffern}}}

, mit

n\in \mathbb {N} ^{+}

.

Zudem lässt sich auch eine rekursive Definition angeben:

R_{n}={\begin{cases}1,&{\text{wenn }}n=1{\text{,}}\\10^{n-1}+R_{n-1},&{\text{wenn }}n\geq 2{\text{.}}\end{cases}}

Die Zahl $R_{n}$ besteht also aus genau $n$ Einsen ( $n\geq 1$ ). Die Folge der Repunits beginnt wie folgt: 1, 11, 111, 1111, … (Folge A002275 in OEIS).

Repunit-Primzahlen

Die Definition der Repunits entstand historisch auf der Suche nach einer Zerlegung solcher Zahlen in ihre Primfaktoren. Die Frage, ob eine Repunit-Zahl eine Primzahl ist, beschäftigte Mathematiker schon im 19. Jahrhundert. So verfasste Carl Gustav Jacob Jacobi eine Arbeit mit dem Titel „Untersuchung, ob die Zahl 11111111111 eine Primzahl ist oder nicht. Ein Kuriosum, veranlasst durch Dase.“

Es ist einfach zu zeigen, dass $R_{n}$ durch $R_{a}$ teilbar ist, falls $n$ durch $a$ teilbar ist. Zum Beispiel ist $R_{9}$ teilbar durch $R_{3}$ : 111111111 = 111 · 1001001. Deshalb muss notwendig $n$ eine Primzahl sein, damit $R_{n}$ eine Primzahl sein kann. Diese Bedingung ist jedoch nicht hinreichend, zum Beispiel ist $R_{3}$ keine Primzahl, da $R_{3}=111=3\cdot 37$ .

Außer für dieses Beispiel von $R_{3}$ kann $p$ nur Teiler von $R_{n}$ sein (für eine Primzahl $n$ ), wenn $p=2kn+1$ für ein bestimmtes $k$ .

Repunit-Primzahlen sind selten. $R_{n}$ ist eine Primzahl für $n=2,19,23,317,1031,\ldots$ (Folge A004023 in OEIS). Die im September 1999 von Harvey Dubner bzw. im Oktober 2000 von Lew Baxter gefundenen $R_{49081}$ und $R_{86453}$ sind wahrscheinlich Primzahlen. Ende März 2007 ermittelten Paul Bourdelais und Harvey Dubner $R_{109297}$ als primzahlverdächtig, vier Monate später fanden Maksym Voznyy und Anton Budnyy $R_{270343}$ als gegenwärtig (2013) größte bekannte wahrscheinliche Repunit-Primzahl.^[2] Es wird vermutet, dass es unendlich viele Repunit-Primzahlen gibt.^[3]

Verallgemeinerte Repunits

Da die obige Definition von Repunits auf dem Dezimalsystem beruht, mag diese Definition zunächst willkürlich erscheinen. Man kann die zugrunde liegende Idee jedoch verallgemeinern, indem man Repunits bezüglich einer beliebigen Basis $b$ definiert:

R_{n}^{(b)}={\frac {b^{n}-1}{b-1}}=\sum _{k=0}^{n-1}b^{k}=\underbrace {\overbrace {11\dotso 1} ^{n{\text{ Ziffern}}}{}_{b}} _{{\text{Zahl zur Basis }}b}

, mit

b,n\in \mathbb {N}

,

b\geq 2

,

n\geq 1

Die verallgemeinerte rekursive Definition lautet:

R_{n}^{(b)}={\begin{cases}1,&{\text{wenn }}n=1{\text{,}}\\b^{n-1}+R_{n-1}^{(b)},&{\text{wenn }}n\geq 2{\text{.}}\end{cases}}

Die Zahl $R_{n}^{(b)}$ besteht also aus genau $n$ Einsen ( $n\geq 1$ ), wenn sie als Zahl zur Basis $b$ notiert wird (wobei $R_{1}^{(b)}$ unabhängig von der Basis immer gleich 1 ist).

Wertetabelle einiger Repunits als Beispiel:

Gegenüberstellung einiger Repunit-Werte $R_{n}^{(b)}$ für gebräuchliche Zahlensysteme
$R_{n}^{(b)}$	Binärsystem $b=2$		Oktalsystem $b=8$		Dezimalsystem $b=10$	Hexadezimalsystem $b=16$
$n$	binär	dezimal	oktal	dezimal	dezimal	hexadezimal	dezimal
1	1₂	1₁₀	1₈	1₁₀	1₁₀	1₁₆	1₁₀
2	11₂	3₁₀	11₈	9₁₀	11₁₀	11₁₆	17₁₀
3	111₂	7₁₀	111₈	73₁₀	111₁₀	111₁₆	273₁₀
4	1111₂	15₁₀	1111₈	585₁₀	1111₁₀	1111₁₆	4369₁₀
5	11111₂	31₁₀	11111₈	4681₁₀	11111₁₀	11111₁₆	69905₁₀
6	111111₂	63₁₀	1111118	37449₁₀	111111₁₀	111111₁₆	1118481₁₀
7	1111111₂	127₁₀	1111111₈	299593₁₀	1111111₁₀	1111111₁₆	17895697₁₀
8	11111111₂	255₁₀	11111111₈	2396745₁₀	11111111₁₀	11111111₁₆	286331153₁₀
9	111111111₂	511₁₀	111111111₈	19173961₁₀	111111111₁₀	111111111₁₆	4581298449₁₀
10	1111111111₂	1023₁₀	1111111111₈	153391689₁₀	1111111111₁₀	1111111111₁₆	73300775185₁₀

Es ist einfach zu beweisen, dass für jedes $n$ , das nicht ohne Rest durch 2 oder $p$ teilbar ist, eine Repunit zur Basis $2p$ existiert, die ein Vielfaches von $n$ ist.

Die Basis-2-Repunits sind bekannt als die Mersenne-Zahlen: $M_{n}=2^{n}-1$

Die Repunit-Primzahlen sind eine Teilmenge der permutierbaren Primzahlen, also der Primzahlen, die Primzahlen bleiben, wenn man ihre Ziffern beliebig vertauscht.

Eine besonders große verallgemeinerte Repunit-Primzahl mit 37.090 Stellen berechnete Andy Steward 2006 mit $\textstyle {\frac {28839^{8317}-1}{28838}}$ . Im Jahr 2010 fand Tom Wu mit $\textstyle {\frac {1549^{12973}-1}{1548}}$ eine noch größere mit 41.382 Stellen.^[4] Die derzeit (August 2020) größte bekannte verallgemeinerte Repunit-Primzahl ist ${\frac {7176^{24691}-1}{7175}}$ mit 95.202 Stellen.^[5]

Repunit-Primzahl zu unterschiedlichen Basen

Beispiele:

Die Repunit $R_{7}^{(5)}=1111111_{5}$ ist zur Basis $b=5$ eine Primzahl, weil $1111111_{5}={\underline {1}}\cdot 5^{6}+{\underline {1}}\cdot 5^{5}+{\underline {1}}\cdot 5^{4}+{\underline {1}}\cdot 5^{3}+{\underline {1}}\cdot 5^{2}+{\underline {1}}\cdot 5^{1}+{\underline {1}}\cdot 5^{0}=15625+3125+625+125+25+5+1=19531\in \mathbb {P}$ eine Primzahl ist.
Es folgt eine Tabelle der kleinsten Repunit-Primzahlen $R_{n}^{(b)}$ zu Basen $b\leq 12$ , im Dezimalsystem geschrieben

Bases $b$ such that $R_{p}(b)$ is prime for prime $p$

Smallest base $b$ such that $R_{p}(b)$ is prime (where $p$ is the $n$ th prime) are

2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39, 12, 22, 67, 120, 195, 48, 54, 463, 38, 41, 17, 808, 404, 46, 76, 793, 38, 28, 215, 37, 236, 59, 15, 514, 260, 498, 6, 2, 95, 3, ... Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS

Smallest base $b$ such that $R_{p}(-b)$ is prime (where $p$ is the $n$ th prime) are

3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, 159, 10, 16, 209, 2, 16, 23, 273, 2, 460, 22, 3, 36, 28, 329, 43, 69, 86, 271, 396, 28, 83, 302, 209, 11, 300, 159, 79, 31, 331, 52, 176, 3, 28, 217, 14, 410, 252, 718, 164, ... Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS

$p$	positive bases $b$ such that $R_{p}(b)$ is prime	negative bases $b$ such that $R_{p}(b)$ is prime
2	2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, 306, 310, 312, 316, 330, 336, 346, 348, 352, 358, 366, 372, 378, 382, 388, 396, 400, 408, 418, 420, 430, 432, 438, 442, 448, 456, 460, 462, 466, 478, 486, 490, 498, 502, 508, 520, 522, 540, 546, 556, 562, 568, 570, 576, 586, 592, 598, 600, 606, 612, 616, 618, 630, 640, 642, 646, 652, 658, 660, 672, 676, 682, 690, 700, 708, 718, 726, 732, 738, 742, 750, 756, 760, 768, 772, 786, 796, 808, 810, 820, 822, 826, 828, 838, 852, 856, 858, 862, 876, 880, 882, 886, 906, 910, 918, 928, 936, 940, 946, 952, 966, 970, 976, 982, 990, 996, 1008, 1012, 1018, 1020, 1030, 1032, 1038, 1048, 1050, 1060, 1062, 1068, 1086, 1090, 1092, 1096, 1102, 1108, 1116, 1122, 1128, 1150, 1152, 1162, 1170, 1180, 1186, 1192, 1200, 1212, 1216, 1222, 1228, 1230, 1236, 1248, 1258, 1276, 1278, 1282, 1288, 1290, 1296, 1300, 1302, 1306, 1318, 1320, 1326, 1360, 1366, 1372, 1380, 1398, 1408, 1422, 1426, 1428, 1432, 1438, 1446, 1450, 1452, 1458, 1470, 1480, 1482, 1486, 1488, 1492, 1498, 1510, 1522, 1530, 1542, 1548, 1552, 1558, 1566, 1570, 1578, 1582, 1596, 1600, 1606, 1608, 1612, 1618, 1620, 1626, 1636, 1656, 1662, 1666, 1668, 1692, 1696, 1698, 1708, 1720, 1722, 1732, 1740, 1746, 1752, 1758, 1776, 1782, 1786, 1788, 1800, 1810, 1822, 1830, 1846, 1860, 1866, 1870, 1872, 1876, 1878, 1888, 1900, 1906, 1912, 1930, 1932, 1948, 1950, 1972, 1978, 1986, 1992, 1996, 1998, 2002, 2010, 2016, 2026, 2028, 2038, 2052, 2062, 2068, 2080, 2082, 2086, 2088, 2098, 2110, 2112, 2128, 2130, 2136, 2140, 2142, 2152, 2160, 2178, 2202, 2206, 2212, 2220, 2236, 2238, 2242, 2250, 2266, 2268, 2272, 2280, 2286, 2292, 2296, 2308, 2310, 2332, 2338, 2340, 2346, 2350, 2356, 2370, 2376, 2380, 2382, 2388, 2392, 2398, 2410, 2416, 2422, 2436, 2440, 2446, 2458, 2466, 2472, 2476, ...	3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, 308, 312, 314, 318, 332, 338, 348, 350, 354, 360, 368, 374, 380, 384, 390, 398, 402, 410, 420, 422, 432, 434, 440, 444, 450, 458, 462, 464, 468, 480, 488, 492, 500, 504, 510, 522, 524, 542, 548, 558, 564, 570, 572, 578, 588, 594, 600, 602, 608, 614, 618, 620, 632, 642, 644, 648, 654, 660, 662, 674, 678, 684, 692, 702, 710, 720, 728, 734, 740, 744, 752, 758, 762, 770, 774, 788, 798, 810, 812, 822, 824, 828, 830, 840, 854, 858, 860, 864, 878, 882, 884, 888, 908, 912, 920, 930, 938, 942, 948, 954, 968, 972, 978, 984, 992, 998, 1010, 1014, 1020, 1022, 1032, 1034, 1040, 1050, 1052, 1062, 1064, 1070, 1088, 1092, 1094, 1098, 1104, 1110, 1118, 1124, 1130, 1152, 1154, 1164, 1172, 1182, 1188, 1194, 1202, 1214, 1218, 1224, 1230, 1232, 1238, 1250, 1260, 1278, 1280, 1284, 1290, 1292, 1298, 1302, 1304, 1308, 1320, 1322, 1328, 1362, 1368, 1374, 1382, 1400, 1410, 1424, 1428, 1430, 1434, 1440, 1448, 1452, 1454, 1460, 1472, 1482, 1484, 1488, 1490, 1494, 1500, 1512, 1524, 1532, 1544, 1550, 1554, 1560, 1568, 1572, 1580, 1584, 1598, 1602, 1608, 1610, 1614, 1620, 1622, 1628, 1638, 1658, 1664, 1668, 1670, 1694, 1698, 1700, 1710, 1722, 1724, 1734, 1742, 1748, 1754, 1760, 1778, 1784, 1788, 1790, 1802, 1812, 1824, 1832, 1848, 1862, 1868, 1872, 1874, 1878, 1880, 1890, 1902, 1908, 1914, 1932, 1934, 1950, 1952, 1974, 1980, 1988, 1994, 1998, 2000, 2004, 2012, 2018, 2028, 2030, 2040, 2054, 2064, 2070, 2082, 2084, 2088, 2090, 2100, 2112, 2114, 2130, 2132, 2138, 2142, 2144, 2154, 2162, 2180, 2204, 2208, 2214, 2222, 2238, 2240, 2244, 2252, 2268, 2270, 2274, 2282, 2288, 2294, 2298, 2310, 2312, 2334, 2340, 2342, 2348, 2352, 2358, 2372, 2378, 2382, 2384, 2390, 2394, 2400, 2412, 2418, 2424, 2438, 2442, 2448, 2460, 2468, 2474, 2478, ...
3	2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278, 279, 287, 288, 290, 293, 309, 314, 329, 332, 336, 342, 344, 348, 351, 357, 369, 378, 381, 383, 392, 395, 398, 402, 404, 405, 414, 416, 426, 434, 435, 447, 453, 455, 456, 476, 489, 495, 500, 512, 518, 525, 530, 531, 533, 537, 540, 551, 554, 560, 566, 567, 572, 579, 582, 584, 603, 605, 609, 612, 621, 624, 626, 635, 642, 644, 668, 671, 677, 686, 696, 701, 720, 726, 728, 735, 743, 747, 755, 761, 762, 768, 773, 782, 785, 792, 798, 801, 812, 818, 819, 825, 827, 836, 839, 846, 855, 857, 860, 864, 875, 878, 890, 894, 897, 899, 911, 915, 918, 920, 927, 950, 959, 960, 969, 974, 981, 987, 990, 992, 993, 1001, 1002, 1007, 1011, 1016, 1020, 1022, 1029, 1041, 1058, 1065, 1067, 1070, 1074, 1077, 1091, 1092, 1097, 1098, 1107, 1116, 1118, 1125, 1130, 1133, 1142, 1146, 1148, 1149, 1163, 1181, 1182, 1191, 1193, 1196, 1200, 1202, 1209, 1217, 1221, 1230, 1254, 1256, 1260, 1263, 1272, 1274, 1275, 1281, 1295, 1308, 1314, 1317, 1323, 1331, 1340, 1343, 1349, 1352, 1359, 1364, 1365, 1373, 1380, 1386, 1392, 1401, 1415, 1421, 1422, 1427, 1434, 1440, 1442, 1443, 1448, 1449, 1454, 1473, 1475, 1484, 1487, 1494, 1505, 1515, 1520, 1526, 1529, 1536, 1539, 1547, 1548, 1559, 1560, 1562, 1566, 1568, 1581, 1583, 1590, 1592, 1596, 1611, 1616, 1620, 1632, 1644, 1646, 1650, 1658, 1659, 1665, 1676, 1685, 1692, 1695, 1700, 1701, 1707, 1709, 1716, 1718, 1746, 1748, 1757, 1767, 1770, 1788, 1800, 1806, 1809, 1811, 1814, 1818, 1826, 1832, 1841, 1847, 1848, 1853, 1863, 1886, 1895, 1902, 1923, 1931, 1932, 1952, 1956, 1970, 1973, 1974, 1977, 1982, 1986, 1994, 2000, 2010, 2015, 2016, 2022, 2033, 2043, 2049, 2054, 2058, 2064, 2073, 2075, 2079, 2085, 2091, 2105, 2112, 2117, 2129, 2136, 2138, 2142, 2145, 2157, 2159, 2162, 2163, 2168, 2171, 2183, 2201, 2204, 2205, 2238, 2241, 2247, 2255, 2259, 2262, 2264, 2273, 2276, 2280, 2301, 2303, 2309, 2313, 2318, 2324, 2339, 2345, 2360, 2364, 2372, 2373, 2385, 2387, 2390, 2397, 2402, 2409, 2411, 2429, 2430, 2435, 2442, 2450, 2451, 2456, 2463, 2465, 2471, 2478, 2483, 2484, 2493, 2495, 2498, ...	2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 21, 22, 25, 28, 34, 39, 42, 51, 55, 58, 60, 63, 67, 70, 72, 76, 78, 79, 81, 90, 91, 100, 102, 106, 111, 112, 118, 120, 132, 139, 142, 144, 148, 151, 154, 156, 162, 163, 165, 168, 169, 174, 177, 189, 190, 193, 195, 204, 207, 210, 216, 219, 232, 237, 246, 247, 267, 273, 279, 280, 288, 289, 291, 294, 310, 315, 330, 333, 337, 343, 345, 349, 352, 358, 370, 379, 382, 384, 393, 396, 399, 403, 405, 406, 415, 417, 427, 435, 436, 448, 454, 456, 457, 477, 490, 496, 501, 513, 519, 526, 531, 532, 534, 538, 541, 552, 555, 561, 567, 568, 573, 580, 583, 585, 604, 606, 610, 613, 622, 625, 627, 636, 643, 645, 669, 672, 678, 687, 697, 702, 721, 727, 729, 736, 744, 748, 756, 762, 763, 769, 774, 783, 786, 793, 799, 802, 813, 819, 820, 826, 828, 837, 840, 847, 856, 858, 861, 865, 876, 879, 891, 895, 898, 900, 912, 916, 919, 921, 928, 951, 960, 961, 970, 975, 982, 988, 991, 993, 994, 1002, 1003, 1008, 1012, 1017, 1021, 1023, 1030, 1042, 1059, 1066, 1068, 1071, 1075, 1078, 1092, 1093, 1098, 1099, 1108, 1117, 1119, 1126, 1131, 1134, 1143, 1147, 1149, 1150, 1164, 1182, 1183, 1192, 1194, 1197, 1201, 1203, 1210, 1218, 1222, 1231, 1255, 1257, 1261, 1264, 1273, 1275, 1276, 1282, 1296, 1309, 1315, 1318, 1324, 1332, 1341, 1344, 1350, 1353, 1360, 1365, 1366, 1374, 1381, 1387, 1393, 1402, 1416, 1422, 1423, 1428, 1435, 1441, 1443, 1444, 1449, 1450, 1455, 1474, 1476, 1485, 1488, 1495, 1506, 1516, 1521, 1527, 1530, 1537, 1540, 1548, 1549, 1560, 1561, 1563, 1567, 1569, 1582, 1584, 1591, 1593, 1597, 1612, 1617, 1621, 1633, 1645, 1647, 1651, 1659, 1660, 1666, 1677, 1686, 1693, 1696, 1701, 1702, 1708, 1710, 1717, 1719, 1747, 1749, 1758, 1768, 1771, 1789, 1801, 1807, 1810, 1812, 1815, 1819, 1827, 1833, 1842, 1848, 1849, 1854, 1864, 1887, 1896, 1903, 1924, 1932, 1933, 1953, 1957, 1971, 1974, 1975, 1978, 1983, 1987, 1995, 2001, 2011, 2016, 2017, 2023, 2034, 2044, 2050, 2055, 2059, 2065, 2074, 2076, 2080, 2086, 2092, 2106, 2113, 2118, 2130, 2137, 2139, 2143, 2146, 2158, 2160, 2163, 2164, 2169, 2172, 2184, 2202, 2205, 2206, 2239, 2242, 2248, 2256, 2260, 2263, 2265, 2274, 2277, 2281, 2302, 2304, 2310, 2314, 2319, 2325, 2340, 2346, 2361, 2365, 2373, 2374, 2386, 2388, 2391, 2398, 2403, 2410, 2412, 2430, 2431, 2436, 2443, 2451, 2452, 2457, 2464, 2466, 2472, 2479, 2484, 2485, 2494, 2496, 2499, ...
5	2, 7, 12, 13, 17, 22, 23, 24, 28, 29, 30, 40, 43, 44, 50, 62, 63, 68, 73, 74, 77, 79, 83, 85, 94, 99, 110, 117, 118, 120, 122, 127, 129, 134, 143, 145, 154, 162, 164, 165, 172, 175, 177, 193, 198, 204, 208, 222, 227, 239, 249, 254, 255, 260, 263, 265, 274, 275, 277, 285, 288, 292, 304, 308, 327, 337, 340, 352, 359, 369, 373, 393, 397, 408, 414, 417, 418, 437, 439, 448, 457, 459, 474, 479, 490, 492, 495, 503, 505, 514, 519, 528, 530, 538, 539, 540, 550, 557, 563, 567, 568, 572, 579, 594, 604, 617, 637, 645, 650, 662, 679, 694, 699, 714, 728, 745, 750, 765, 770, 772, 793, 804, 805, 824, 837, 854, 860, 864, 868, 880, 890, 919, 942, 954, 967, 968, 974, 979, 1007, 1009, 1044, 1045, 1047, 1053, 1078, 1079, 1085, 1102, 1108, 1129, 1133, 1135, 1152, 1154, 1157, 1173, 1178, 1183, 1187, 1209, 1228, 1239, 1249, 1253, 1282, 1284, 1297, 1327, 1333, 1343, 1348, 1352, 1370, 1377, 1382, 1398, 1419, 1429, 1452, 1462, 1463, 1475, 1480, 1493, 1503, 1520, 1525, 1553, 1557, 1562, 1563, 1579, 1657, 1662, 1663, 1668, 1672, 1679, 1683, 1684, 1694, 1695, 1704, 1748, 1749, 1755, 1760, 1762, 1767, 1770, 1788, 1838, 1843, 1859, 1883, 1914, 1927, 1944, 1947, 1960, 1975, 1977, 1979, 1987, 1990, 2032, 2034, 2037, 2052, 2059, 2069, 2080, 2090, 2097, 2098, 2118, 2123, 2134, 2164, 2167, 2175, 2177, 2218, 2228, 2233, 2243, 2245, 2252, 2272, 2274, 2289, 2295, 2299, 2300, 2305, 2318, 2328, 2332, 2340, 2362, 2373, 2375, 2382, 2383, 2387, 2430, 2432, 2438, 2448, 2450, 2464, 2483, 2487, 2499, ...	2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411, 427, 438, 452, 462, 483, 487, 495, 500, 505, 506, 507, 510, 528, 531, 551, 555, 583, 588, 603, 606, 610, 628, 641, 642, 652, 658, 676, 693, 702, 707, 716, 738, 741, 758, 760, 768, 782, 791, 792, 812, 813, 815, 836, 850, 861, 901, 912, 935, 960, 973, 978, 980, 982, 1012, 1023, 1026, 1035, 1045, 1057, 1065, 1066, 1068, 1071, 1087, 1093, 1105, 1116, 1127, 1137, 1148, 1155, 1156, 1158, 1160, 1165, 1166, 1178, 1187, 1191, 1215, 1230, 1243, 1265, 1266, 1270, 1276, 1281, 1291, 1310, 1320, 1341, 1342, 1353, 1365, 1368, 1375, 1401, 1433, 1452, 1462, 1473, 1475, 1488, 1496, 1506, 1530, 1541, 1550, 1551, 1560, 1567, 1585, 1593, 1598, 1605, 1611, 1615, 1618, 1621, 1637, 1643, 1660, 1661, 1676, 1686, 1692, 1695, 1698, 1715, 1716, 1717, 1738, 1752, 1761, 1775, 1780, 1791, 1805, 1818, 1831, 1838, 1840, 1842, 1846, 1848, 1873, 1901, 1903, 1923, 1926, 1928, 1935, 1940, 1946, 1947, 1948, 1962, 1970, 1990, 1996, 2017, 2023, 2025, 2046, 2058, 2066, 2080, 2102, 2111, 2112, 2143, 2160, 2165, 2170, 2192, 2210, 2222, 2227, 2232, 2233, 2256, 2258, 2265, 2271, 2275, 2293, 2303, 2320, 2326, 2330, 2332, 2333, 2335, 2343, 2353, 2366, 2386, 2390, 2392, 2440, 2457, 2463, 2473, ...
7	2, 3, 5, 6, 13, 14, 17, 26, 31, 38, 40, 46, 56, 60, 61, 66, 68, 72, 73, 80, 87, 89, 93, 95, 115, 122, 126, 128, 146, 149, 156, 158, 160, 163, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 251, 264, 266, 280, 282, 290, 294, 300, 303, 320, 341, 349, 350, 353, 363, 381, 395, 399, 404, 405, 417, 418, 436, 438, 447, 450, 461, 464, 466, 478, 523, 531, 539, 548, 560, 583, 584, 591, 599, 609, 611, 622, 646, 647, 655, 657, 660, 681, 698, 700, 710, 717, 734, 760, 765, 776, 798, 800, 802, 805, 822, 842, 856, 863, 870, 878, 899, 912, 913, 926, 927, 931, 940, 941, 942, 947, 959, 984, 998, 1029, 1046, 1052, 1055, 1081, 1084, 1087, 1111, 1115, 1117, 1131, 1136, 1144, 1148, 1161, 1169, 1188, 1189, 1194, 1195, 1197, 1221, 1224, 1229, 1248, 1266, 1277, 1280, 1281, 1284, 1291, 1297, 1309, 1316, 1333, 1339, 1348, 1363, 1364, 1371, 1377, 1382, 1402, 1404, 1409, 1418, 1419, 1424, 1425, 1438, 1439, 1455, 1469, 1480, 1482, 1489, 1494, 1512, 1529, 1537, 1547, 1550, 1554, 1558, 1563, 1565, 1566, 1567, 1572, 1579, 1585, 1596, 1605, 1606, 1608, 1609, 1622, 1627, 1630, 1651, 1668, 1672, 1679, 1687, 1700, 1708, 1710, 1714, 1738, 1750, 1753, 1757, 1762, 1766, 1783, 1790, 1796, 1797, 1802, 1808, 1816, 1831, 1839, 1840, 1854, 1855, 1885, 1888, 1899, 1904, 1911, 1915, 1918, 1920, 1929, 1941, 1958, 1978, 1985, 1990, 1998, 2015, 2039, 2057, 2071, 2072, 2097, 2100, 2126, 2127, 2131, 2139, 2151, 2152, 2156, 2158, 2159, 2181, 2186, 2188, 2190, 2193, 2201, 2205, 2219, 2222, 2226, 2230, 2232, 2244, 2245, 2265, 2274, 2293, 2299, 2303, 2306, 2308, 2321, 2359, 2378, 2404, 2408, 2436, 2440, 2450, 2463, 2466, 2480, 2491, 2496, ...	2, 3, 10, 11, 14, 15, 16, 17, 18, 21, 24, 25, 29, 37, 43, 44, 46, 49, 52, 54, 61, 66, 72, 73, 78, 84, 86, 87, 99, 101, 106, 114, 115, 128, 133, 135, 136, 143, 145, 148, 164, 169, 170, 173, 200, 219, 224, 226, 228, 231, 234, 240, 248, 255, 262, 275, 281, 282, 298, 301, 305, 320, 322, 327, 330, 339, 347, 359, 368, 369, 385, 387, 401, 417, 435, 442, 450, 455, 458, 488, 494, 508, 519, 521, 553, 579, 582, 590, 616, 621, 625, 628, 630, 638, 654, 669, 675, 691, 694, 711, 712, 714, 719, 724, 725, 728, 732, 737, 750, 756, 757, 765, 768, 773, 774, 778, 780, 793, 803, 810, 827, 828, 831, 842, 848, 865, 869, 901, 910, 919, 922, 926, 931, 936, 938, 960, 967, 971, 982, 1002, 1010, 1022, 1032, 1060, 1073, 1074, 1089, 1109, 1110, 1123, 1128, 1130, 1131, 1132, 1138, 1160, 1178, 1179, 1184, 1190, 1207, 1216, 1218, 1220, 1221, 1291, 1293, 1302, 1303, 1332, 1349, 1353, 1361, 1362, 1366, 1373, 1374, 1386, 1394, 1407, 1435, 1474, 1487, 1515, 1524, 1535, 1538, 1558, 1562, 1563, 1564, 1597, 1605, 1614, 1615, 1626, 1638, 1639, 1645, 1648, 1653, 1654, 1671, 1681, 1683, 1696, 1698, 1705, 1710, 1713, 1719, 1723, 1726, 1731, 1739, 1748, 1758, 1768, 1781, 1787, 1789, 1793, 1799, 1800, 1810, 1817, 1823, 1830, 1835, 1839, 1848, 1853, 1859, 1879, 1909, 1929, 1932, 1935, 1961, 1964, 1969, 1979, 1996, 1997, 2002, 2009, 2013, 2026, 2037, 2056, 2058, 2074, 2075, 2077, 2080, 2088, 2104, 2111, 2133, 2140, 2145, 2147, 2174, 2175, 2196, 2207, 2216, 2219, 2221, 2233, 2234, 2236, 2248, 2251, 2257, 2269, 2273, 2277, 2285, 2286, 2294, 2305, 2315, 2319, 2322, 2334, 2339, 2356, 2378, 2403, 2406, 2418, 2419, 2424, 2425, 2427, 2432, 2434, 2446, 2460, 2467, 2479, 2480, 2482, 2489, 2494, ...
11	5, 17, 20, 21, 30, 53, 60, 86, 137, 172, 195, 212, 224, 229, 258, 268, 272, 319, 339, 355, 365, 366, 389, 390, 398, 414, 467, 480, 504, 534, 539, 543, 567, 592, 619, 626, 654, 709, 735, 756, 766, 770, 778, 787, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1091, 1103, 1194, 1210, 1213, 1239, 1243, 1249, 1264, 1309, 1311, 1341, 1348, 1357, 1377, 1425, 1433, 1437, 1448, 1449, 1459, 1510, 1528, 1575, 1586, 1601, 1631, 1661, 1696, 1698, 1701, 1748, 1770, 1840, 1859, 1864, 1874, 1910, 1924, 1942, 1954, 1977, 1985, 1993, 2000, 2029, 2039, 2087, 2089, 2123, 2149, 2167, 2215, 2228, 2238, 2261, 2274, 2294, 2301, 2307, 2319, 2320, 2323, 2338, 2393, 2403, 2416, 2438, 2457, 2458, 2461, ...	2, 6, 12, 13, 23, 24, 26, 35, 62, 69, 91, 105, 147, 160, 163, 183, 185, 193, 229, 232, 233, 236, 248, 262, 269, 280, 294, 303, 315, 330, 376, 394, 426, 430, 440, 464, 469, 491, 492, 508, 537, 625, 629, 647, 653, 666, 752, 772, 775, 786, 788, 832, 840, 852, 854, 855, 859, 905, 922, 972, 993, 998, 1012, 1041, 1043, 1061, 1076, 1082, 1129, 1136, 1158, 1173, 1177, 1199, 1208, 1237, 1266, 1277, 1312, 1336, 1338, 1370, 1386, 1405, 1439, 1442, 1455, 1508, 1522, 1543, 1553, 1564, 1566, 1626, 1665, 1734, 1775, 1810, 1821, 1829, 1840, 1844, 1908, 1910, 1938, 1963, 1982, 2009, 2017, 2040, 2070, 2083, 2101, 2105, 2118, 2163, 2170, 2201, 2203, 2224, 2231, 2244, 2329, 2346, 2347, 2373, 2433, 2450, 2456, 2461, 2486, 2493, ...
13	2, 3, 5, 7, 34, 37, 43, 59, 72, 94, 98, 110, 133, 149, 151, 159, 190, 207, 219, 221, 251, 260, 264, 267, 282, 286, 291, 319, 355, 363, 373, 382, 397, 398, 402, 406, 408, 412, 436, 442, 486, 489, 507, 542, 544, 552, 553, 582, 585, 592, 603, 610, 614, 634, 643, 645, 689, 708, 720, 730, 744, 769, 772, 806, 851, 853, 862, 882, 912, 928, 930, 952, 968, 993, 1006, 1019, 1025, 1040, 1047, 1097, 1098, 1120, 1125, 1131, 1133, 1153, 1165, 1172, 1175, 1191, 1245, 1248, 1271, 1273, 1281, 1299, 1305, 1306, 1311, 1346, 1376, 1378, 1400, 1403, 1404, 1412, 1428, 1442, 1450, 1471, 1502, 1570, 1585, 1660, 1662, 1666, 1736, 1754, 1757, 1836, 1853, 1861, 1885, 1903, 1913, 1915, 1972, 1975, 1987, 1988, 1995, 2006, 2033, 2052, 2073, 2098, 2191, 2193, 2206, 2210, 2252, 2255, 2267, 2280, 2314, 2319, 2324, 2346, 2362, 2366, 2417, 2444, 2450, 2479, 2486, ...	2, 3, 21, 22, 23, 35, 39, 74, 80, 84, 89, 108, 114, 121, 126, 134, 152, 153, 171, 180, 195, 204, 230, 256, 263, 297, 321, 326, 336, 342, 345, 351, 368, 390, 393, 397, 398, 399, 413, 427, 439, 490, 525, 563, 566, 574, 591, 602, 609, 630, 641, 652, 657, 660, 667, 682, 685, 694, 708, 712, 716, 721, 755, 760, 763, 791, 797, 813, 828, 839, 850, 862, 882, 898, 999, 1025, 1029, 1030, 1035, 1038, 1049, 1063, 1073, 1076, 1081, 1105, 1111, 1128, 1162, 1176, 1178, 1185, 1216, 1227, 1238, 1264, 1268, 1274, 1307, 1318, 1324, 1359, 1369, 1371, 1373, 1388, 1392, 1445, 1450, 1451, 1461, 1489, 1496, 1529, 1539, 1550, 1555, 1560, 1610, 1612, 1616, 1632, 1695, 1794, 1805, 1824, 1857, 1873, 1889, 1899, 1930, 1943, 1950, 1955, 1963, 1979, 1985, 1994, 2003, 2005, 2010, 2085, 2103, 2108, 2113, 2122, 2151, 2153, 2172, 2189, 2214, 2228, 2234, 2236, 2257, 2268, 2293, 2307, 2314, 2337, 2373, 2380, 2400, 2433, 2436, 2483, 2487, 2492, ...
17	2, 11, 20, 21, 28, 31, 55, 57, 62, 84, 87, 97, 107, 109, 129, 147, 149, 157, 160, 170, 181, 189, 191, 207, 241, 247, 251, 274, 295, 297, 315, 327, 335, 349, 351, 355, 364, 365, 368, 379, 383, 410, 419, 423, 431, 436, 438, 466, 472, 506, 513, 527, 557, 571, 597, 599, 614, 637, 653, 656, 688, 708, 709, 720, 740, 762, 835, 836, 874, 974, 976, 980, 982, 986, 1029, 1045, 1054, 1136, 1151, 1171, 1181, 1185, 1222, 1230, 1236, 1240, 1280, 1319, 1332, 1367, 1371, 1375, 1392, 1402, 1435, 1447, 1481, 1495, 1496, 1519, 1525, 1530, 1536, 1547, 1556, 1592, 1603, 1607, 1615, 1683, 1692, 1706, 1726, 1775, 1776, 1782, 1793, 1828, 1833, 1846, 1945, 1967, 1988, 1998, 2001, 2002, 2046, 2049, 2064, 2100, 2103, 2108, 2165, 2183, 2192, 2206, 2223, 2249, 2259, 2268, 2295, 2301, 2303, 2312, 2321, 2343, 2351, 2371, 2395, 2406, 2443, 2446, 2468, 2471, 2492, 2494, ...	2, 7, 13, 17, 19, 41, 48, 58, 59, 66, 86, 129, 133, 139, 143, 146, 149, 166, 167, 231, 268, 270, 299, 328, 359, 387, 397, 408, 469, 523, 527, 534, 541, 553, 555, 569, 582, 583, 600, 608, 634, 664, 667, 672, 673, 709, 714, 720, 725, 733, 746, 759, 776, 802, 808, 822, 860, 870, 877, 892, 896, 902, 911, 962, 970, 975, 1034, 1050, 1051, 1082, 1090, 1109, 1148, 1150, 1168, 1185, 1198, 1200, 1209, 1212, 1233, 1235, 1261, 1281, 1312, 1338, 1339, 1357, 1380, 1382, 1387, 1403, 1411, 1440, 1441, 1459, 1477, 1483, 1497, 1503, 1560, 1561, 1578, 1583, 1689, 1691, 1710, 1751, 1752, 1755, 1789, 1797, 1826, 1883, 1887, 1901, 1912, 1950, 1955, 1965, 1986, 2016, 2032, 2038, 2041, 2060, 2071, 2074, 2083, 2093, 2121, 2126, 2139, 2180, 2182, 2193, 2195, 2247, 2292, 2340, 2461, 2493, ...
19	2, 10, 11, 12, 14, 19, 24, 40, 45, 46, 48, 65, 66, 67, 75, 85, 90, 103, 105, 117, 119, 137, 147, 164, 167, 179, 181, 205, 220, 235, 242, 253, 254, 263, 268, 277, 303, 315, 332, 337, 366, 369, 370, 389, 399, 404, 424, 431, 446, 449, 480, 481, 506, 509, 521, 523, 531, 547, 567, 573, 581, 622, 646, 651, 673, 736, 768, 787, 797, 807, 810, 811, 817, 840, 846, 857, 867, 869, 870, 888, 899, 902, 971, 988, 990, 992, 1010, 1023, 1030, 1041, 1042, 1072, 1102, 1112, 1116, 1143, 1147, 1208, 1218, 1223, 1252, 1256, 1257, 1260, 1291, 1330, 1335, 1340, 1402, 1454, 1536, 1538, 1594, 1622, 1625, 1631, 1642, 1648, 1693, 1701, 1709, 1726, 1734, 1759, 1767, 1779, 1794, 1809, 1822, 1858, 1954, 1988, 1992, 1999, 2024, 2030, 2065, 2067, 2081, 2101, 2135, 2150, 2180, 2193, 2196, 2219, 2239, 2293, 2325, 2330, 2336, 2358, 2382, 2390, 2402, 2403, 2409, 2457, 2461, 2478, 2480, 2499, ...	2, 10, 13, 24, 28, 43, 47, 49, 54, 65, 83, 98, 106, 143, 152, 177, 184, 190, 194, 195, 241, 249, 259, 264, 292, 315, 319, 345, 353, 355, 386, 394, 481, 500, 517, 525, 534, 535, 556, 595, 601, 649, 656, 680, 686, 687, 697, 707, 710, 756, 798, 804, 817, 818, 829, 839, 841, 858, 864, 891, 906, 912, 932, 948, 973, 991, 994, 1012, 1030, 1032, 1036, 1046, 1057, 1060, 1066, 1071, 1089, 1166, 1175, 1203, 1205, 1227, 1237, 1261, 1283, 1284, 1319, 1327, 1335, 1336, 1337, 1356, 1372, 1387, 1412, 1413, 1439, 1450, 1460, 1483, 1488, 1495, 1502, 1515, 1518, 1521, 1532, 1676, 1677, 1685, 1686, 1720, 1722, 1734, 1739, 1758, 1759, 1762, 1782, 1792, 1814, 1822, 1833, 1834, 1838, 1839, 1869, 1884, 1922, 1972, 1999, 2005, 2012, 2033, 2042, 2056, 2093, 2119, 2221, 2244, 2251, 2264, 2325, 2356, 2359, 2371, 2381, 2383, 2427, 2436, 2446, 2479, ...
23	10, 40, 82, 113, 127, 141, 170, 257, 275, 287, 295, 315, 344, 373, 442, 468, 609, 634, 646, 663, 671, 710, 819, 834, 857, 884, 894, 904, 992, 997, 1060, 1069, 1077, 1120, 1143, 1190, 1232, 1253, 1261, 1280, 1291, 1347, 1407, 1436, 1448, 1483, 1514, 1570, 1642, 1684, 1691, 1744, 1786, 1801, 1816, 1830, 1879, 1891, 1906, 1946, 1968, 1994, 2005, 2158, 2200, 2318, 2444, 2459, ...	2, 3, 7, 16, 17, 18, 25, 46, 47, 106, 110, 111, 118, 136, 144, 145, 230, 238, 361, 382, 422, 439, 474, 494, 495, 519, 588, 639, 657, 707, 733, 751, 802, 831, 863, 902, 925, 976, 994, 1001, 1059, 1140, 1181, 1240, 1249, 1319, 1375, 1442, 1458, 1508, 1699, 1734, 1751, 1757, 1760, 1766, 1807, 1849, 1897, 1904, 1914, 1922, 1978, 2029, 2057, 2075, 2139, 2166, 2183, 2209, 2216, 2241, 2246, 2270, 2307, 2309, 2367, 2429, 2446, 2456, ...
29	6, 40, 65, 70, 114, 151, 221, 229, 268, 283, 398, 451, 460, 519, 554, 587, 627, 628, 659, 687, 699, 859, 884, 915, 943, 974, 986, 1101, 1120, 1159, 1176, 1212, 1223, 1297, 1312, 1322, 1337, 1390, 1409, 1415, 1446, 1477, 1508, 1592, 1636, 1691, 1800, 1802, 1803, 1820, 2020, 2126, 2414, 2421, 2429, 2461, ...	7, 15, 25, 62, 119, 123, 154, 245, 285, 294, 295, 357, 371, 476, 626, 664, 690, 708, 723, 737, 768, 783, 803, 825, 826, 835, 841, 842, 867, 871, 897, 904, 934, 953, 1066, 1069, 1088, 1097, 1108, 1183, 1197, 1202, 1259, 1302, 1364, 1461, 1497, 1528, 1559, 1638, 1715, 1760, 1764, 1815, 1821, 1849, 1854, 1874, 2047, 2086, 2091, 2100, 2133, 2160, 2231, 2242, 2316, 2422, ...
31	2, 14, 19, 31, 44, 53, 71, 82, 117, 127, 131, 145, 177, 197, 203, 241, 258, 261, 276, 283, 293, 320, 325, 379, 387, 388, 406, 413, 461, 462, 470, 486, 491, 534, 549, 569, 582, 612, 618, 639, 696, 706, 723, 746, 765, 767, 774, 796, 802, 877, 878, 903, 923, 981, 991, 998, 1002, 1008, 1049, 1077, 1113, 1114, 1115, 1138, 1146, 1159, 1177, 1189, 1200, 1229, 1270, 1281, 1312, 1359, 1429, 1486, 1493, 1496, 1558, 1605, 1614, 1626, 1632, 1649, 1673, 1698, 1754, 1790, 1859, 1865, 1894, 1900, 1904, 1918, 1939, 1948, 1973, 1998, 1999, 2022, 2031, 2038, 2055, 2111, 2167, 2169, 2180, 2236, 2300, 2377, 2387, 2422, 2462, 2473, ...	2, 6, 10, 36, 65, 74, 78, 83, 106, 115, 120, 148, 161, 163, 168, 176, 189, 194, 197, 266, 270, 288, 331, 385, 399, 407, 410, 412, 413, 431, 468, 513, 524, 546, 569, 572, 578, 581, 600, 611, 625, 626, 647, 719, 723, 756, 832, 834, 849, 922, 986, 1006, 1007, 1047, 1064, 1102, 1199, 1217, 1247, 1335, 1349, 1353, 1372, 1374, 1382, 1400, 1405, 1410, 1418, 1419, 1422, 1426, 1444, 1457, 1554, 1568, 1590, 1622, 1678, 1680, 1683, 1684, 1695, 1725, 1840, 1847, 1861, 1874, 1895, 1939, 1950, 1987, 1988, 1990, 1999, 2070, 2081, 2093, 2180, 2183, 2212, 2216, 2220, 2227, 2266, 2303, 2337, 2349, 2389, 2408, 2418, 2426, ...
37	61, 77, 94, 97, 99, 113, 126, 130, 134, 147, 161, 172, 187, 202, 208, 246, 261, 273, 285, 302, 320, 432, 444, 503, 523, 525, 563, 666, 680, 709, 740, 757, 787, 902, 962, 964, 969, 1013, 1109, 1165, 1186, 1219, 1351, 1528, 1587, 1660, 1698, 1710, 1754, 1811, 1972, 2070, 2088, 2137, 2145, 2185, 2210, 2305, 2348, 2365, 2396, 2398, 2439, 2453, 2462, ...	16, 19, 21, 49, 56, 63, 71, 74, 77, 83, 92, 96, 99, 160, 172, 197, 198, 230, 241, 280, 283, 415, 425, 448, 490, 520, 627, 691, 735, 784, 803, 829, 842, 853, 871, 872, 893, 894, 973, 981, 989, 1043, 1060, 1061, 1071, 1179, 1182, 1203, 1290, 1299, 1317, 1370, 1389, 1477, 1478, 1496, 1562, 1601, 1624, 1714, 1789, 1848, 1890, 1942, 2016, 2083, 2156, 2217, 2250, 2253, 2266, 2313, 2315, 2319, 2323, 2343, 2358, 2361, 2363, 2383, 2396, 2398, 2399, 2401, 2486, ...
41	14, 53, 55, 58, 71, 76, 82, 211, 248, 271, 296, 316, 430, 433, 439, 472, 545, 553, 555, 596, 663, 677, 682, 746, 814, 832, 885, 926, 947, 959, 1048, 1121, 1146, 1175, 1207, 1300, 1302, 1390, 1404, 1445, 1466, 1541, 1650, 1699, 1702, 1706, 1756, 1757, 1782, 1902, 1944, 1963, 1994, 2012, 2042, 2122, 2163, 2344, 2348, 2369, 2439, 2442, 2465, ...	61, 63, 99, 144, 230, 312, 360, 401, 413, 424, 451, 515, 542, 567, 610, 618, 622, 651, 690, 732, 817, 871, 1007, 1100, 1156, 1278, 1403, 1427, 1460, 1535, 1572, 1604, 1681, 1742, 1802, 1820, 1847, 1903, 1910, 1913, 1978, 2019, 2104, 2134, 2152, 2169, 2309, 2383, 2491, ...
43	15, 21, 26, 86, 89, 114, 123, 163, 180, 310, 332, 377, 409, 438, 448, 457, 477, 526, 534, 556, 586, 612, 653, 665, 690, 692, 709, 760, 783, 803, 821, 848, 877, 899, 909, 942, 981, 1041, 1042, 1043, 1066, 1068, 1069, 1106, 1126, 1139, 1197, 1310, 1386, 1476, 1561, 1599, 1606, 1636, 1774, 1802, 1835, 1867, 2033, 2068, 2153, 2155, 2229, 2256, 2269, 2276, 2324, 2467, ...	2, 3, 6, 22, 59, 83, 91, 95, 120, 148, 195, 196, 201, 247, 252, 264, 315, 360, 378, 458, 555, 680, 792, 893, 1025, 1088, 1158, 1171, 1240, 1280, 1416, 1437, 1632, 1661, 1677, 1681, 1849, 1946, 1960, 2007, 2090, 2092, 2225, 2242, 2244, 2377, 2483, ...
47	5, 17, 19, 55, 62, 75, 89, 98, 99, 132, 172, 186, 197, 220, 268, 278, 279, 288, 439, 443, 496, 579, 583, 587, 742, 777, 825, 911, 966, 1003, 1029, 1155, 1202, 1267, 1377, 1461, 1472, 1473, 1530, 1562, 1634, 1682, 1705, 1783, 1848, 1852, 1885, 1900, 1943, 1957, 1969, 2036, 2044, 2105, 2111, 2265, 2282, 2485, ...	6, 7, 17, 90, 126, 139, 143, 257, 293, 295, 319, 387, 482, 519, 603, 720, 819, 884, 896, 903, 905, 921, 952, 954, 956, 1042, 1058, 1147, 1170, 1237, 1253, 1279, 1295, 1343, 1366, 1370, 1406, 1465, 1514, 1593, 1595, 1607, 1609, 1622, 1701, 1705, 1709, 1736, 1772, 1791, 1833, 1860, 1866, 1969, 1990, 1992, 2059, 2115, 2132, 2169, 2219, 2233, 2253, 2344, ...
53	24, 45, 60, 165, 235, 272, 285, 298, 307, 381, 416, 429, 623, 799, 858, 924, 929, 936, 1034, 1067, 1076, 1130, 1223, 1243, 1327, 1488, 1592, 1594, 1659, 1673, 1676, 1678, 1762, 1873, 1917, 1928, 1997, 2110, 2117, 2141, 2166, 2231, 2243, 2293, 2338, 2380, 2420, 2424, 2428, 2463, 2466, ...	10, 14, 40, 57, 111, 119, 406, 447, 475, 620, 646, 839, 848, 866, 909, 997, 1086, 1095, 1180, 1318, 1319, 1332, 1418, 1447, 1472, 1534, 1617, 1681, 1684, 1735, 1788, 1955, 2037, 2118, 2120, 2163, 2169, 2170, 2390, 2407, 2440, 2498, ...
59	19, 70, 102, 116, 126, 188, 209, 257, 294, 359, 451, 461, 468, 470, 638, 653, 710, 762, 766, 781, 824, 901, 939, 964, 995, 1036, 1047, 1098, 1150, 1211, 1234, 1243, 1256, 1278, 1389, 1401, 1422, 1436, 1454, 1492, 1523, 1639, 1703, 1705, 1826, 1913, 1915, 1978, 2013, 2043, 2131, 2140, 2200, 2208, 2228, 2229, 2307, 2309, 2323, 2365, 2480, 2485, ...	6, 9, 25, 46, 89, 92, 109, 133, 136, 140, 167, 173, 213, 239, 255, 277, 337, 350, 359, 553, 554, 586, 594, 599, 639, 692, 710, 815, 860, 864, 1015, 1030, 1050, 1094, 1106, 1110, 1112, 1195, 1199, 1211, 1216, 1260, 1347, 1363, 1370, 1459, 1476, 1477, 1507, 1541, 1556, 1597, 1599, 1620, 1636, 1662, 1696, 1724, 1757, 1786, 1860, 1867, 1923, 1949, 1984, 2004, 2081, 2116, 2177, 2219, 2231, 2305, 2376, 2417, 2443, 2494, ...
61	2, 19, 69, 88, 138, 155, 205, 234, 336, 420, 425, 455, 470, 525, 555, 561, 608, 626, 667, 674, 766, 779, 846, 851, 937, 971, 998, 1053, 1055, 1112, 1137, 1263, 1329, 1383, 1445, 1448, 1499, 1544, 1551, 1578, 1640, 1689, 1724, 1750, 1793, 1868, 2143, 2241, 2358, 2407, 2414, ...	2, 7, 70, 178, 208, 251, 274, 276, 290, 326, 328, 350, 413, 452, 552, 558, 594, 595, 605, 607, 626, 787, 791, 801, 905, 971, 1019, 1091, 1117, 1140, 1198, 1241, 1274, 1357, 1428, 1462, 1604, 1647, 1654, 1705, 1717, 1908, 1987, 2061, 2109, 2161, 2309, 2372, 2450, 2491, ...
67	46, 122, 238, 304, 314, 315, 328, 332, 346, 372, 382, 426, 440, 491, 496, 510, 524, 528, 566, 638, 733, 826, 1016, 1054, 1071, 1214, 1309, 1338, 1388, 1401, 1457, 1512, 1536, 1582, 1624, 1718, 1773, 1814, 1816, 1825, 1952, 1985, 2021, 2072, 2308, 2349, 2449, 2481, ...	5, 10, 23, 33, 40, 54, 193, 326, 330, 364, 375, 382, 388, 404, 438, 449, 562, 625, 626, 683, 700, 765, 797, 807, 1001, 1017, 1136, 1181, 1216, 1242, 1249, 1254, 1286, 1386, 1412, 1482, 1581, 1656, 1748, 1832, 1873, 1921, 2017, 2038, 2061, 2166, 2193, 2204, 2253, 2271, 2418, 2475, ...
71	3, 6, 17, 24, 37, 89, 132, 374, 387, 402, 421, 435, 453, 464, 490, 516, 708, 736, 919, 947, 981, 1033, 1067, 1170, 1195, 1253, 1284, 1349, 1385, 1409, 1479, 1709, 1724, 1726, 1735, 1875, 1950, 1984, 2012, 2070, 2124, 2133, 2161, 2194, 2424, 2432, 2459, ...	46, 94, 99, 189, 226, 236, 244, 372, 387, 390, 409, 410, 424, 442, 478, 540, 574, 608, 611, 644, 653, 695, 707, 846, 868, 1036, 1248, 1336, 1374, 1395, 1418, 1424, 1549, 1665, 1737, 1856, 1866, 1880, 1917, 1937, 2105, 2114, 2126, 2141, 2166, 2202, 2217, 2274, 2297, 2298, 2311, 2314, 2321, 2377, 2430, 2431, ...
73	11, 15, 75, 114, 195, 215, 295, 335, 378, 559, 566, 650, 660, 832, 871, 904, 966, 1021, 1112, 1203, 1334, 1433, 1485, 1724, 1822, 1959, 1998, 2115, 2154, 2432, 2465, 2486, ...	18, 214, 280, 394, 422, 444, 447, 571, 745, 787, 796, 886, 954, 960, 987, 1012, 1055, 1140, 1194, 1212, 1224, 1227, 1349, 1583, 1598, 1640, 1686, 1714, 1723, 1750, 1931, 1962, 2032, 2036, 2110, 2223, 2339, ...
79	22, 112, 140, 158, 170, 254, 271, 330, 334, 354, 390, 483, 528, 560, 565, 714, 850, 888, 924, 929, 933, 935, 970, 1019, 1047, 1141, 1266, 1338, 1359, 1376, 1412, 1485, 1504, 1542, 1598, 1607, 1618, 1747, 1773, 1814, 1843, 2087, 2088, 2100, 2167, 2186, 2233, 2311, 2358, 2359, 2363, 2429, ...	2, 20, 22, 35, 47, 72, 109, 133, 184, 211, 226, 259, 352, 470, 559, 720, 785, 800, 823, 842, 895, 1003, 1145, 1172, 1213, 1291, 1318, 1375, 1441, 1453, 1460, 1461, 1467, 1477, 1604, 1608, 1637, 1654, 1695, 1703, 1807, 1831, 1834, 1903, 1948, 2035, 2060, 2065, 2072, 2081, 2335, 2348, 2393, ...
83	41, 146, 386, 593, 667, 688, 906, 927, 930, 1025, 1032, 1111, 1410, 1437, 1638, 1829, 1960, 2045, 2381, 2384, ...	49, 75, 458, 471, 634, 734, 798, 809, 932, 1139, 1268, 1400, 1498, 1963, 1989, 2112, 2177, 2233, 2252, 2349, 2365, 2446, ...
89	2, 114, 159, 190, 234, 251, 436, 616, 834, 878, 1008, 1049, 1060, 1062, 1118, 1472, 1689, 1792, 2282, 2334, 2463, 2494, ...	16, 20, 93, 195, 227, 325, 465, 758, 888, 911, 1075, 1301, 1590, 1640, 1783, 1807, 2168, 2204, 2231, 2376, ...
97	12, 90, 104, 234, 271, 339, 420, 421, 428, 429, 464, 805, 909, 934, 1054, 1114, 1116, 1128, 1144, 1159, 1193, 1364, 1788, 2086, 2215, 2254, 2448, 2461, ...	70, 121, 300, 317, 348, 404, 412, 460, 515, 605, 839, 843, 904, 953, 1130, 1148, 1342, 1466, 1674, 1779, 1855, 2080, 2108, 2193, 2466, ...
101	22, 78, 164, 302, 332, 359, 387, 428, 456, 564, 617, 697, 703, 704, 785, 831, 979, 1003, 1045, 1327, 1388, 1400, 1505, 1586, 1845, 1855, 1969, 1990, 2015, 2082, 2104, 2274, 2339, ...	2, 5, 22, 98, 110, 131, 132, 137, 197, 206, 212, 333, 353, 371, 372, 441, 449, 500, 509, 658, 703, 711, 748, 786, 925, 961, 971, 1025, 1217, 1278, 1298, 1382, 1414, 1511, 1672, 1766, 1832, 1937, 2053, 2188, 2198, 2367, 2480, ...
103	3, 52, 345, 392, 421, 472, 584, 617, 633, 761, 767, 775, 785, 839, 1033, 1496, 1594, 1660, 1673, 1692, 1694, 1891, 1940, 2001, 2011, 2075, 2105, 2108, 2261, 2292, 2331, 2339, 2347, 2372, 2396, ...	5, 45, 96, 107, 138, 143, 164, 223, 249, 272, 339, 425, 453, 497, 506, 601, 703, 739, 750, 780, 859, 932, 942, 1010, 1037, 1173, 1274, 1291, 1357, 1399, 1482, 1491, 1552, 1602, 1791, 2206, 2390, 2482, 2491, ...
107	2, 19, 61, 68, 112, 157, 219, 349, 677, 692, 700, 809, 823, 867, 999, 1177, 1210, 1237, 1405, 1471, 1532, 1577, 1578, 1660, 1664, 1715, 1751, 2194, 2232, 2287, 2342, 2373, 2407, 2458, 2493, ...	6, 22, 46, 56, 79, 357, 377, 458, 469, 640, 678, 849, 980, 1064, 1106, 1156, 1194, 1208, 1338, 1449, 1483, 1666, 1743, 1753, 1796, 1803, 1841, 2034, 2280, 2321, ...
109	12, 57, 72, 79, 89, 129, 158, 165, 239, 240, 260, 277, 313, 342, 421, 445, 577, 945, 1003, 1122, 1198, 1224, 1268, 1381, 1399, 1428, 1504, 1506, 1565, 1567, 1638, 1674, 1784, 1968, 2312, 2365, ...	12, 23, 26, 31, 74, 118, 169, 192, 261, 503, 564, 568, 602, 655, 673, 725, 874, 885, 969, 972, 995, 1097, 1114, 1192, 1201, 1219, 1257, 1568, 1664, 1686, 1784, 1840, 1865, 1878, 1952, 1969, 1972, 1993, 2016, 2064, 2211, 2218, 2241, 2245, 2295, 2421, ...
113	86, 233, 266, 299, 334, 492, 592, 641, 656, 719, 946, 1404, 1477, 1511, 1602, 1635, 1669, 1824, 1833, 1962, 2169, 2241, 2448, 2498, 2499, ...	92, 116, 473, 636, 759, 822, 871, 982, 998, 1041, 1576, 1814, 1915, 1938, ...
127	2, 5, 6, 47, 50, 126, 151, 226, 250, 401, 427, 473, 477, 486, 497, 585, 624, 644, 678, 685, 687, 758, 896, 897, 936, 1076, 1399, 1509, 1690, 1782, 1830, 1837, 1842, 1938, 1983, 2009, 2196, 2241, 2285, 2324, ...	2, 35, 333, 368, 391, 422, 436, 554, 641, 674, 721, 847, 874, 1357, 1516, 1604, 1672, 1762, 1886, 1950, 2028, 2153, 2219, 2322, 2405, 2411, ...
131	7, 493, 567, 591, 593, 613, 764, 883, 899, 919, 953, 1219, 1283, 1322, 1343, 1368, 1772, 1968, 2482, ...	48, 96, 281, 420, 577, 806, 1055, 1121, 1124, 1419, 2003, 2442, ...
137	13, 166, 213, 355, 586, 669, 707, 768, 833, 1003, 1105, 1189, 1202, 1314, 1383, 1526, 1594, 1640, 1949, 1999, 2120, 2428, ...	89, 242, 287, 367, 527, 599, 681, 731, 795, 802, 1072, 1162, 1329, 1556, 1589, 1612, 1754, 1779, 2194, 2394, 2397, ...
139	11, 50, 221, 415, 521, 577, 580, 668, 717, 720, 738, 902, 1389, 1392, 1581, 1650, 1731, 1765, 1913, 1941, 1944, 2220, 2306, 2378, ...	30, 84, 285, 294, 554, 629, 710, 739, 775, 951, 1023, 1161, 1372, 1480, 1515, 1541, 1612, 1653, 1755, 1837, 1849, 2067, 2221, 2313, ...
149	5, 7, 68, 79, 106, 260, 319, 502, 550, 779, 855, 1067, 1385, 1423, 1434, 1595, 1688, 1693, 1723, 1735, 1873, 1954, 1993, 2098, 2099, 2438, ...	16, 39, 51, 105, 171, 184, 221, 236, 273, 338, 370, 380, 630, 776, 927, 990, 1248, 1294, 1427, 1431, 1471, 1628, 1866, 1933, 2219, 2275, ...
151	29, 55, 57, 160, 176, 222, 255, 364, 427, 439, 642, 660, 697, 863, 1066, 1124, 1148, 1207, 1309, 1365, 1382, 1480, 1486, 2076, 2145, 2241, 2270, 2273, ...	147, 198, 334, 368, 371, 516, 727, 1114, 1216, 1324, 1360, 1417, 1526, 1557, 1570, 1699, 1748, 1785, 1805, 2117, 2182, 2342, 2378, ...
157	56, 71, 76, 181, 190, 317, 338, 413, 426, 609, 694, 794, 797, 960, 1008, 1035, 1071, 1165, 1285, 1294, 1301, 1306, 1404, 1483, 1549, 1560, 1787, 2227, 2281, 2474, ...	19, 33, 45, 132, 133, 157, 214, 281, 495, 679, 749, 777, 792, 993, 1205, 1400, 1489, 1631, 1673, 1772, 1850, 1906, 2053, 2056, 2071, 2086, 2215, 2433, 2458, ...
163	30, 62, 118, 139, 147, 291, 456, 755, 834, 888, 902, 924, 1033, 1086, 1664, 1700, 1840, 1924, 1969, 1970, 2111, 2196, 2278, ...	19, 52, 126, 148, 197, 248, 258, 374, 386, 542, 551, 644, 688, 737, 805, 878, 928, 1134, 1145, 1156, 1237, 1398, 1426, 1492, 1516, 1720, 2145, 2410, ...
167	44, 45, 127, 175, 182, 403, 449, 453, 476, 571, 582, 700, 749, 764, 929, 957, 1048, 1259, 1433, 1563, 1599, 1602, 1908, 2085, 2159, 2489, ...	2, 38, 62, 85, 178, 259, 278, 360, 419, 583, 735, 860, 883, 925, 934, 1073, 1202, 1227, 1558, 1591, 1705, 1742, 1788, 1871, 1953, 1966, 2101, 2226, 2234, 2385, 2483, ...
173	60, 62, 139, 141, 303, 313, 368, 425, 542, 663, 1147, 1219, 1407, 1453, 1699, 1734, 2232, 2347, 2417, ...	16, 30, 281, 326, 390, 695, 732, 815, 1338, 1675, 1718, 1822, 2163, 2219, ...
179	304, 478, 586, 942, 952, 975, 1138, 1858, 2003, 2284, 2498, ...	11, 55, 255, 289, 321, 467, 869, 1230, 1435, 1447, 1587, 1927, 2024, 2160, 2234, 2475, ...
181	5, 37, 171, 427, 509, 571, 618, 665, 671, 786, 1028, 1187, 1542, 1677, 2044, 2180, 2440, ...	289, 536, 584, 632, 656, 670, 1012, 1025, 1042, 1133, 1288, 1461, 1605, 1651, 1720, 1945, 2121, 2143, 2212, 2242, 2482, 2484, ...
191	74, 214, 416, 477, 595, 664, 699, 712, 743, 924, 1148, 1238, 1742, 1877, 2038, 2056, 2195, 2292, 2446, ...	2, 36, 76, 527, 575, 602, 1136, 1206, 1488, 1495, 1644, 1875, 2111, ...
193	118, 301, 486, 554, 637, 673, 736, 1074, 1198, 1568, 1767, 1824, 1938, 1970, 2200, 2275, 2322, ...	12, 408, 495, 606, 657, 916, 1013, 1117, 1179, 1237, 1528, 1558, 1732, 1742, 1909, 2117, 2456, ...
197	33, 236, 248, 262, 335, 363, 388, 593, 763, 813, 1038, 1053, 1348, 1358, 1426, 1543, 1736, 1925, 1977, 1985, 2036, 2152, 2179, 2250, 2331, ...	52, 54, 192, 387, 441, 639, 814, 1239, 1350, 1441, 1514, 1542, 1614, 1674, 1678, 1742, 1862, 2180, 2307, 2332, ...
199	156, 362, 383, 401, 442, 630, 645, 689, 740, 921, 936, 944, 983, 988, 1120, 1189, 1208, 1642, 1785, 2134, 2213, ...	2, 136, 207, 307, 328, 424, 467, 489, 669, 749, 808, 830, 1096, 1255, 1262, 1341, 1607, 1631, 1642, 1871, 1882, 1885, 1895, 2149, 2153, 2423, ...
211	46, 57, 354, 478, 539, 581, 653, 829, 835, 977, 1084, 1109, 1148, 1318, 1375, 1693, 1821, 1875, 1981, 2026, 2145, 2268, 2278, 2407, 2454, ...	66, 69, 290, 642, 661, 764, 841, 959, 982, 1173, 1337, 1456, 1515, 1590, 1680, 1703, 1780, 1792, 2292, 2463, ...
223	183, 186, 219, 221, 661, 749, 905, 914, 1027, 1362, 1400, 1622, 1840, 1847, 2069, 2104, 2181, ...	9, 46, 52, 108, 235, 361, 508, 591, 699, 736, 746, 781, 910, 924, 928, 1014, 1078, 1200, 1345, 1506, 1550, 1879, 1932, 2024, 2182, 2376, 2449, ...
227	72, 136, 235, 240, 251, 322, 350, 500, 523, 556, 577, 671, 688, 743, 967, 1119, 1138, 1207, 1218, 1340, 1447, 1519, 1551, 1645, 1783, 1905, 2093, 2492, ...	22, 26, 57, 80, 129, 178, 246, 401, 502, 832, 926, 1009, 1011, 1162, 1301, 1313, 1321, 1348, 1352, 1354, 1419, 1884, 1948, 2122, 2154, 2155, 2162, 2182, 2298, 2316, 2490, ...
229	606, 725, 754, 858, 950, 1582, 1605, 1749, 1897, 2225, 2263, 2314, 2323, 2418, 2451, 2488, ...	5, 11, 55, 101, 202, 444, 472, 843, 1012, 1016, 1065, 1129, 1551, 1588, 1626, 1741, 2312, 2332, 2447, ...
233	602, 1394, 1397, 2266, ...	489, 786, 1092, 1167, 1524, 1677, 1917, 1923, ...
239	223, 260, 367, 474, 564, 862, 1431, 1740, ...	69, 126, 213, 490, 693, 845, 1507, 1579, 1655, 2302, ...
241	115, 163, 223, 265, 270, 330, 689, 849, 1014, 1030, 1431, 1564, 1827, 1980, 2090, 2306, 2419, ...	137, 218, 346, 622, 679, 694, 768, 886, 948, 1018, 1080, 1640, 1722, 1762, 1790, 1802, 1899, 2023, 2156, 2200, ...
251	37, 246, 267, 618, 933, 1122, 2371, ...	16, 43, 219, 308, 443, 567, 648, 747, 811, 1732, 1762, 1848, 2395, 2474, ...
257	52, 78, 435, 459, 658, 709, 1022, 1027, 1197, 1216, 1254, 1275, 1378, 1490, 1660, 1680, 1954, 2282, 2351, 2494, ...	36, 171, 259, 368, 484, 487, 676, 1056, 1199, 1249, 1303, 1593, 1637, 1950, 2269, 2441, 2454, ...
263	104, 131, 161, 476, 494, 563, 735, 842, 909, 987, 1091, 1120, 1540, 1777, 1908, 2032, 2108, 2148, 2211, 2399, ...	96, 120, 154, 255, 267, 307, 400, 453, 633, 661, 792, 946, 1169, 1329, 1381, 1444, 1619, 1814, 1843, 1886, 2043, 2070, 2303, ...
269	41, 48, 294, 493, 520, 812, 843, 1163, 1751, 1753, 1902, ...	76, 176, 547, 623, 682, 756, 842, 1028, 1048, 1087, 1130, 1132, 1198, 1245, 1489, 1641, 1655, 1718, 1761, 1835, 2205, ...
271	6, 21, 186, 201, 222, 240, 586, 622, 624, 1088, 1205, 1248, 1253, 1401, 1469, 1503, 1782, 1862, 1864, 2121, 2198, 2203, 2235, 2279, 2369, ...	117, 411, 446, 630, 806, 816, 905, 1001, 1372, 1709, 1789, 1935, 2120, ...
277	338, 473, 637, 940, 941, 978, 1605, 2002, 2467, 2485, ...	26, 43, 72, 200, 215, 378, 418, 591, 840, 889, 955, 968, 1699, 1830, 1869, ...
281	217, 446, 606, 618, 790, 864, 1556, 1718, ...	3, 145, 549, 794, 1017, 1284, 1894, ...
283	13, 197, 254, 288, 323, 374, 404, 943, 1108, 1363, 1493, 1732, 1950, 1978, 2375, 2396, 2408, 2465, ...	159, 172, 232, 322, 323, 370, 387, 463, 647, 727, 866, 952, 975, 991, 1054, 1095, 1464, 1746, 1776, 1878, 2232, 2315, 2418, 2440, ...
293	136, 388, 471, 1283, 1772, 1916, 2420, ...	10, 82, 100, 115, 122, 485, 899, 1135, 1457, 1791, 2161, ...

List of repunit primes base $b$

Smallest prime $p>2$ such that $R_{p}(b)$ is prime are (start with $b=2$ , 0 if no such $p$ exists)

3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3, 4421, 439, 7, 5, 7, 3343, 17, 13, 3, 0, ... Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS

Smallest prime $p>2$ such that $R_{p}(-b)$ is prime are (start with $b=2$ , 0 if no such $p$ exists)

3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, 7, 13, 5, 3, 37, 3, 3, 5, 3, 293, 19, 7, 167, 7, 7, 709, 13, 3, 3, 37, 89, 71, 43, 37, (>500000), 19, 7, 3, ... Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS

$b$	numbers $n$ such that $R_{n}(b)$ is prime (some large terms are only corresponding to probable primes, these $n$ are checked up to ≥200000)	OEIS sequence
−50	1153, 26903, 56597, ...	Vorlage:OEIS link
−49	7, 19, 37, 83, 1481, 12527, 20149, ...	Vorlage:OEIS link
−48	2^*, 5, 17, 131, 84589, ...	Vorlage:OEIS link
−47	5, 19, 23, 79, 1783, 7681, ...	Vorlage:OEIS link
−46	7, 23, 59, 71, 107, 223, 331, 2207, 6841, 94841, ...	Vorlage:OEIS link
−45	103, 157, 37159, ...	Vorlage:OEIS link
−44	2^*, 7, 41233, ...	Vorlage:OEIS link
−43	5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573, ...	Vorlage:OEIS link
−42	2^*, 3, 709, 1637, 17911, 127609, 172663, ...	Vorlage:OEIS link
−41	17, 691, 113749, ...	Vorlage:OEIS link
−40	53, 67, 1217, 5867, 6143, 11681, 29959, ...	Vorlage:OEIS link
−39	3, 13, 149, 15377, ...	Vorlage:OEIS link
−38	2^*, 5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591, ...	Vorlage:OEIS link
−37	5, 7, 2707, 163193, ...	Vorlage:OEIS link
−36	31, 191, 257, 367, 3061, 110503, ...	Vorlage:OEIS link
−35	11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, 135623, ...	Vorlage:OEIS link
−34	3, 294277, ...
−33	5, 67, 157, 12211, ...	Vorlage:OEIS link
−32	2^* (no others)
−31	109, 461, 1061, 50777, ...	Vorlage:OEIS link
−30	2^*, 139, 173, 547, 829, 2087, 2719, 3109, 10159, 56543, 80599, ...	Vorlage:OEIS link
−29	7, 112153, 151153, ...	Vorlage:OEIS link
−28	3, 19, 373, 419, 491, 1031, 83497, ...	Vorlage:OEIS link
−27	(none)
−26	11, 109, 227, 277, 347, 857, 2297, 9043, ...	Vorlage:OEIS link
−25	3, 7, 23, 29, 59, 1249, 1709, 1823, 1931, 3433, 8863, 43201, 78707, ...	Vorlage:OEIS link
−24	2^*, 7, 11, 19, 2207, 2477, 4951, ...	Vorlage:OEIS link
−23	11, 13, 67, 109, 331, 587, 24071, 29881, 44053, ...	Vorlage:OEIS link
−22	3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287, ...	Vorlage:OEIS link
−21	3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599, 394579, ...	Vorlage:OEIS link
−20	2^*, 5, 79, 89, 709, 797, 1163, 6971, 140053, 177967, 393257, ...	Vorlage:OEIS link
−19	17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951, 476929, ...	Vorlage:OEIS link
−18	2^*, 3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147, ...	Vorlage:OEIS link
−17	7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259, ...	Vorlage:OEIS link
−16	3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393, ...	Vorlage:OEIS link
−15	3, 7, 29, 1091, 2423, 54449, 67489, 551927, ...	Vorlage:OEIS link
−14	2^*, 7, 53, 503, 1229, 22637, 1091401, ...	Vorlage:OEIS link
−13	3, 11, 17, 19, 919, 1151, 2791, 9323, 56333, 1199467, ...	Vorlage:OEIS link
−12	2^*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, 495953, ...	Vorlage:OEIS link
−11	5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ...	Vorlage:OEIS link
−10	5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787, ...	Vorlage:OEIS link
−9	3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, 860029, ...	Vorlage:OEIS link
−8	2^* (no others)
−7	3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, 1178033, ...	Vorlage:OEIS link
−6	2^*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, 1313371, ...	Vorlage:OEIS link
−5	5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, 1856147, ...	Vorlage:OEIS link
−4	2^*, 3 (no others)
−3	2^*, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, 1896463, 2533963, ...	Vorlage:OEIS link
−2	3, 4^*, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ...	Vorlage:OEIS link
2	2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, ..., 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ...	Vorlage:OEIS link
3	3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, ...	Vorlage:OEIS link
4	2 (no others)
5	3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ...	Vorlage:OEIS link
6	2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, ...	Vorlage:OEIS link
7	5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...	Vorlage:OEIS link
8	3 (no others)
9	(none)
10	2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...	Vorlage:OEIS link
11	17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ...	Vorlage:OEIS link
12	2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ...	Vorlage:OEIS link
13	5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089, 1503503, ...	Vorlage:OEIS link
14	3, 7, 19, 31, 41, 2687, 19697, 59693, 67421, 441697, ...	Vorlage:OEIS link
15	3, 43, 73, 487, 2579, 8741, 37441, 89009, 505117, 639833, ...	Vorlage:OEIS link
16	2 (no others)
17	3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509, 1990523, ...	Vorlage:OEIS link
18	2, 25667, 28807, 142031, 157051, 180181, 414269, ...	Vorlage:OEIS link
19	19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359, ...	Vorlage:OEIS link
20	3, 11, 17, 1487, 31013, 48859, 61403, 472709, ...	Vorlage:OEIS link
21	3, 11, 17, 43, 271, 156217, 328129, ...	Vorlage:OEIS link
22	2, 5, 79, 101, 359, 857, 4463, 9029, 27823, ...	Vorlage:OEIS link
23	5, 3181, 61441, 91943, 121949, 221411, ...	Vorlage:OEIS link
24	3, 5, 19, 53, 71, 653, 661, 10343, 49307, 115597, 152783, ...	Vorlage:OEIS link
25	(none)
26	7, 43, 347, 12421, 12473, 26717, ...	Vorlage:OEIS link
27	3 (no others)
28	2, 5, 17, 457, 1423, 115877, ...	Vorlage:OEIS link
29	5, 151, 3719, 49211, 77237, ...	Vorlage:OEIS link
30	2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883, ...	Vorlage:OEIS link
31	7, 17, 31, 5581, 9973, 101111, 535571, ...	Vorlage:OEIS link
32	(none)
33	3, 197, 3581, 6871, 183661, ...	Vorlage:OEIS link
34	13, 1493, 5851, 6379, 125101, ...	Vorlage:OEIS link
35	313, 1297, ...
36	2 (no others)
37	13, 71, 181, 251, 463, 521, 7321, 36473, 48157, 87421, 168527, 249341, ...	Vorlage:OEIS link
38	3, 7, 401, 449, 109037, ...	Vorlage:OEIS link
39	349, 631, 4493, 16633, 36341, ...	Vorlage:OEIS link
40	2, 5, 7, 19, 23, 29, 541, 751, 1277, ...	Vorlage:OEIS link
41	3, 83, 269, 409, 1759, 11731, ...	Vorlage:OEIS link
42	2, 1319, ...
43	5, 13, 6277, 26777, 27299, 40031, 44773, 194119, ...	Vorlage:OEIS link
44	5, 31, 167, 100511, ...	Vorlage:OEIS link
45	19, 53, 167, 3319, 11257, 34351, 216551, ...	Vorlage:OEIS link
46	2, 7, 19, 67, 211, 433, 2437, 2719, 19531, ...	Vorlage:OEIS link
47	127, 18013, 39623, ...	Vorlage:OEIS link
48	19, 269, 349, 383, 1303, 15031, 200443, ...	Vorlage:OEIS link
49	(none)
50	3, 5, 127, 139, 347, 661, 2203, 6521, 210319, ...	Vorlage:OEIS link

^* Repunits with negative base and even n are negative. If their absolute value is prime then they are included above and marked with an asterisk. They are not included in the corresponding OEIS sequences.

For more information, see.^[6]^[7]^[8]^[9]

Algebra factorization of generalized repunit numbers

If b is a perfect power (can be written as mⁿ, with m, n integers, n > 1) differs from 1, then there is at most one repunit in base-b. If n is a prime power (can be written as p^r, with p prime, r integer, p, r >0), then all repunit in base-b are not prime aside from R_p and R₂. R_p can be either prime or composite, the former examples, b = −216, −128, 4, 8, 16, 27, 36, 100, 128, 256, etc., the latter examples, b = −243, −125, −64, −32, −27, −8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc., and R₂ can be prime (when p differs from 2) only if b is negative, a power of −2, for example, b = −8, −32, −128, −8192, etc., in fact, the R₂ can also be composite, for example, b = −512, −2048, −32768, etc. If n is not a prime power, then no base-b repunit prime exists, for example, b = 64, 729 (with n = 6), b = 1024 (with n = 10), and b = −1 or 0 (with n any natural number). Another special situation is b = −4k⁴, with k positive integer, which has the aurifeuillean factorization, for example, b = −4 (with k = 1, then R₂ and R₃ are primes), and b = −64, −324, −1024, −2500, −5184, ... (with k = 2, 3, 4, 5, 6, ...), then no base-b repunit prime exists. It is also conjectured that when b is neither a perfect power nor −4k⁴ with k positive integer, then there are infinity many base-b repunit primes.

Weblinks

Eric W. Weisstein: Repunit. In: MathWorld (englisch).
Repunit prime numbers research project.

Einzelnachweise

↑ Albert H. Beiler: Recreations in the Theory of Numbers. The queen of mathematics entertains. 2. Auflage. Dover, New York 1966, Kap. XI, S. 83 ff.
↑ Giovanni Di Maria: The Repunit Primes Project.
↑ Chris K. Caldwell: The Prime Glossery: Repunit.
↑ Andy Steward: Titanic Prime Generalized Repunits. (Memento vom 19. Oktober 2013 im Internet Archive)
↑ Chris K. Caldwell: Generalized Repunit. Abgerufen am 21. August 2020 (englisch).
↑ Repunit primes in base −50 to 50
↑ Repunit primes in base 2 to 160
↑ Repunit primes in base −160 to −2
↑ Repunit primes in base −200 to −2

[1] Albert H. Beiler: Recreations in the Theory of Numbers. The queen of mathematics entertains. 2. Auflage. Dover, New York 1966, Kap. XI, S. 83 ff.

[2] Giovanni Di Maria: The Repunit Primes Project.

[3] Chris K. Caldwell: The Prime Glossery: Repunit.

[4] Andy Steward: Titanic Prime Generalized Repunits. (Memento vom 19. Oktober 2013 im Internet Archive)

[5] Chris K. Caldwell: Generalized Repunit. Abgerufen am 21. August 2020 (englisch).

[6] Repunit primes in base −50 to 50

[7] Repunit primes in base 2 to 160

[8] Repunit primes in base −160 to −2

[9] Repunit primes in base −200 to −2

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[9]

formelbasiert	Carol ((2ⁿ − 1)² − 2) \| Doppelte Mersenne (2^{2^p − 1} − 1) \| Fakultät (n! ± 1) \| Fermat (2^2ⁿ + 1) \| Kubisch (x³ − y³)/(x − y) \| Kynea ((2ⁿ + 1)² − 2) \| Leyland (x^y + y^x) \| Mersenne (2^p − 1) \| Mills (A^3ⁿ) \| Pierpont (2^u⋅3^v + 1) \| Primorial (p_n# ± 1) \| Proth (k⋅2ⁿ + 1) \| Pythagoreisch (4n + 1) \| Quartisch (x⁴ + y⁴) \| Thabit (3⋅2ⁿ − 1) \| Wagstaff ((2^p + 1)/3) \| Williams ((b-1)⋅bⁿ − 1) \| Woodall (n⋅2ⁿ − 1)
Primzahlfolgen	Bell \| Fibonacci \| Lucas \| Motzkin \| Pell \| Perrin
eigenschaftsbasiert	Elitär \| Fortunate \| Gut \| Glücklich \| Higgs \| Hochkototient \| Isoliert \| Pillai \| Ramanujan \| Regulär \| Stark \| Stern \| Wall–Sun–Sun \| Wieferich \| Wilson
basisabhängig	Belphegor \| Champernowne \| Dihedral \| Einzigartig \| Fröhlich \| Keith \| Lange \| Minimal \| Mirp \| Permutierbar \| Primeval \| Palindrom \| Repunit-Primzahl ((10ⁿ − 1)/9) \| Schwach \| Smarandache–Wellin \| Strobogrammatisch \| Tetradisch \| Trunkierbar \| Zirkular
basierend auf Tupel	Ausbalanciert (p − n, p, p + n) \| Chen \| Cousin (p, p + 4) \| Cunningham (p, 2p ± 1, …) \| Drilling (p, p + 2 oder p + 4, p + 6) \| Konstellation \| Sexy (p, p + 6) \| Sichere (p, (p − 1)/2) \| Sophie Germain (p, 2p + 1) \| Vierling (p, p + 2, p + 6, p + 8) \| Zwilling (p, p + 2) \| Zwillings-Bi-Kette (n ± 1, 2n ± 1, …)
nach Größe	Titanisch (1.000+ Stellen) \| Gigantisch (10.000+ Stellen) \| Mega (1.000.000+ Stellen) \| Beva (1.000.000.000+ Stellen)