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
In monotheistic thought, God is usually viewed as the supreme being, creator, and principal object of faith. Swinburne, R.G. "God" in Honderich, Ted. (ed)''The Oxford Companion to Philosophy'', Oxford University Press, 1995. God is typically ...
Osiris
Osiris (, from Egyptian ''wsjr'', cop, ⲟⲩⲥⲓⲣⲉ , ; Phoenician: 𐤀𐤎𐤓, romanized: ʾsr) is the god of fertility, agriculture, the afterlife, the dead, resurrection, life, and vegetation in ancient Egyptian religion. He wa ...
in
Egyptian mythology
Egyptian mythology is the collection of myths from ancient Egypt, which describe the actions of the Egyptian gods as a means of understanding the world around them. The beliefs that these myths express are an important part of ancient Egyp ...
). For example, 132 = 12 + 21 + 13 + 31 + 23 + 32.
[Wells, D. '']The Penguin Dictionary of Curious and Interesting Numbers
''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, ...
'' London: Penguin Group. (1987): 138
Osiris numbers in base ten
In
base ten
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, 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 + 594 + 945 + 954 + 469 + 496 + 649 + 694 + 946 + 964 + 569 + 596 + 659 + 695 + 956 + 965
Maximal Osiris numbers
If
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
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:
:253
9 = 2 + 3 + 5 + 23 + 32 + 25 + 52 + 35 + 53 (base = 9)
::210 = 2 + 3 + 5 + 21 + 29 + 23 + 47 + 32 + 48 (base = 10)
:276
13 = 2 + 6 + 7 + 26 + 62 + 27 + 72 + 67 + 76 (b=13)
::435 = 2 + 6 + 7 + 32 + 80 + 33 + 93 + 85 + 97 (b=10)
:DF53
17 = 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
In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18.
Definition
Let n be a natural number. We define the digit ...
References
{{Classes of natural numbers
Base-dependent integer sequences