Digit-reassembly number

Digit-reassembly numbers, or Osiris numbers, are numbers that are equal to the sum of permutations of sub-samples of their own digits (compare the dismemberment and reconstruction of the god Osiris in Egyptian mythology). For example, 132 = 12 21 13 31 23 32.

Osiris numbers in base ten

In base ten, the smallest Osiris numbers are these, with a number-length of three digits and digit-span of two for the permutated sums:

132 = 12 21 13 31 23 32

264 = 24 42 26 62 46 64

396 = 36 63 39 93 69 96

Note that all are multiples of 132. A larger Osiris number in base ten is this, with a number-length of five digits and digit-span of three for the permutated sums:

35964 = 345 354 435 453 534 543 346 364 436 463 634 643 349 394 439 493 934 943 356 365 536 563 635 653 359 395 539 593 935 953 369 396 639 693 936 963 456 465 546 564 645 654 459 495 549 863 945 954 469 496 649 694 946 964 569 596 659 695 956 965

Maximal Osiris numbers

If zero is treated as a full digit in all positions, then 207 in base ten is a maximal Osiris number, being equal to the sum of all possible distinct numbers formed from permutated sub-samples of its digits:

207 = 2 0 7 20 02 27 72 07 70

In other bases, maximal Osiris numbers exist that do not contain zeros. For example:

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)

Multi-minimal Osiris numbers

Using the same terminology, 132, 264 and 396 are minimal Osiris numbers, being equal to the sums of all numbers formed from permutated samples of only two of their digits. 35964 is also minimal, being the sum of samples of three digits, but 34658 is a multi-minimal Osiris number, being equal to the sums of all numbers formed from permutated samples of one or three of its digits:

34658 = 3 4 5 6 8 345 354 435 453 534 543 346 364 436 463 634 643 348 384 438 483 834 843 356 365 536 563 635 653 358 385 538 583 835 853 368 386 638 683 836 863 456 465 546 564 645 654 458 485 548 584 845 854 468 486 648 684 846 864 568 586 658 685 856 865

30659 and 38657 are similarly multi-minimal, using permutated samples of one and three of their digits.

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

If only numbers with unique non-zero digits are considered, a three-digit number in base ten can have a digit-sum ranging from 6 = 1 2 3 to 24 = 7 8 9. If these potential digit-sums are used in the formula 2 x digit-sum x 11, the digit-sum of the result will determine whether or not the result is an Osiris number.

1. 2 x 6 x 11 = 132.

2. Digit-sum (132) = 1 2 3 = 6.

3. Therefore 132 is an Osiris number.

1. 2 x 7 x 11 = 154.

2. Digit-sum (154) = 1 5 4 = 10.

3. Therefore 154 is not an Osiris number.

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

The test for further five-digit Osiris numbers of the same form (sampling three digits) will use potential digit-sums between 15 = 1 2 3 4 5 and 35 = 5 6 7 8 9. When this range of digit-sums is tested, only 35964 returns the same digit-sum as that used in the formula. These simplified tests considerably reduce the task of finding large Osiris numbers in a particular base. For example, to test by brute force whether permutated six-digit samples of n = 332,639,667,360 are equal to n would involve summing 665,280 numbers, where 665,280 = 12 x 11 x 10 x 9 x 8 x 7 = 12! / 6!. However, because each digit of n occurs 55440 times in each of the six possible positions in the samples, the test is reduced to this:

1. digit-sum (332,639,667,360) = 3 3 2 6 3 9 6 6 7 3 6 0 = 54

2. 55440 x 54 x 111,111 = 332,639,667,360

3. Therefore 332,639,667,360 is an Osiris number.

See also

• Digit sum