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Digit-reassembly number

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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 + 4017 + 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

References

  1. ^ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 138
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