Digit-reassembly number
Digit-reassembly numbers, or Osiris numbers, are numbers that are equal to the sum of permutations of their own digits (compare the dismemberment and reconstruction of the god Osiris in Egyptian mythology). In base 10, the smallest Osiris numbers are these, with a number-length of three and digit-span of two for the permutated sums:
- 132 = 12 + 21 + 13 + 31 + 23 + 32[1]
- 264 = 24 + 42 + 26 + 62 + 46 + 64
- 396 = 36 + 63 + 39 + 93 + 69 + 96
A larger Osiris number in base 10 is this, with a number-length of five and digit-span of three for the permutated sums:
- 35964 = 345 + 354 + 435 + 453 + 534 + 543 + 346 + 364 + 436 + 463 + 634 + 643 + 349 + 394 + 439 + 493 + 934 + 943 + 356 + 365 + 536 + 563 + 635 + 653 + 359 + 395 + 539 + 593 + 935 + 953 + 369 + 396 + 639 + 693 + 936 + 963 + 456 + 465 + 546 + 564 + 645 + 654 + 459 + 495 + 549 + 594 + 945 + 954 + 469 + 496 + 649 + 694 + 946 + 964 + 569 + 596 + 659 + 695 + 956 + 965
Other bases
In other bases, Osiris numbers exist in which the digit-span of permutated sums takes all values from 1 to the number-length minus 1:
- 2539 = 2 + 3 + 5 + 23 + 32 + 25 + 52 + 35 + 53 (base = 9)
- 210 = 2 + 3 + 5 + 21 + 29 + 23 + 47 + 32 + 48 (base = 10)
- 27613 = 2 + 6 + 7 + 26 + 62 + 27 + 72 + 67 + 76 (b=13)
- 435 = 2 + 6 + 7 + 32 + 80 + 33 + 93 + 85 + 97 (b=10)
- DF5317 = 3 + 5 + D + F + 35 + 53 + 3D + D3 + 3F + F3 + 5D + D5 + 5F + F5 + DF + FD + 35D + 3D5 + 53D + 5D3 + D35 + D53 + 35F + 3F5 + 53F + 5F3 + F35 + F53 + 3DF + 3FD + D3F + DF3 + F3D + FD3 + 5DF + 5FD + D5F + DF5 + F5D + FD5 (b=17)
- 68292 = 3 + 5 + 13 + 15 + 56 + 88 + 64 + 224 + 66 + 258 + 98 + 226 + 100 + 260 +
236 + 268 + 965 + 1093 + 1509 + 1669 + 3813 + 3845 + 967 + 1127 + 1511 + 1703 + 4391 + 4423 + 1103 + 1135 + 3823 + 4015 + 4399 + 4559 + 1681 + 1713 + 3857 + 401 7 + 4433 + 4561 (b=10)
Tests for Osiris numbers
Testing for Osiris numbers is simplified when one notes that, for example, each digit of 132 occurs twice in the ones and tens position of the sums:
- 132 = 12 + 21 + 13 + 31 + 23 + 32 = 2x11 + 2x22 + 2x33 = 22 + 44 + 66
The test can be further simplified:
- 132 = 2 x (11 + 22 + 33) = 2 x (1 + 2 + 3) x 11 = 2 x 6 x 11
In 35964, each digit occurs 12 times in the ones, tens and hundreds position of the sums:
- 35964 = 12x333 + 12x444 + 12x555 + 12x666 + 12x999 = 3996 + 5328 + 6660 + 7992 + 11988
- 35964 = 12 x (333 + 444 + 555 + 666 + 999) = 12 x (3 + 4 + 5 + 6 + 9) x 111 = 12 x 27 x 111
See also
- ^ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 138
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