A090056 -id:A090056 - cp's OEIS frontend

A090055 Numbers n divisible by at least one nontrivial permutation (rearrangement) of the digits of n.

105, 108, 405, 510, 540, 702, 703, 810, 1001, 1005, 1008, 1020, 1050, 1053, 1080, 2002, 2016, 2025, 2040, 2050, 2079, 2100, 2106, 3003, 3024, 3042, 3045, 3060, 3105, 3402, 3510, 4004, 4005, 4050, 4070, 4080, 4200, 5005, 5010, 5040

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial permutations are identified as (1) permutation = n, or (2) when n mod 10=0, permutations of n's digits which result in shifting only trailing zeros to the most significant side of n where they drop off, such that permutation = n/10^z, where z <= the number of trailing zeros of n. So if n were 1809000, the following permutations would be excluded as trivial: 1809000, 0180900, 0018090, 0001809.

A031877 (numbers which are multiples of their reversals) and both A084687 and A090053 (numbers divided by number formed by sorting their digits), are subsets of this sequence. This sequence differentiates itself by including terms such as 7425 which is divided by 2475 (a rearrangement of 7425's digits that is neither a reversal or an ascending sort.)

			a(27)=3045 because 3045 is divisible by 435, a nontrivial permutation of 3045. (0435)

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A031877, A084687, A090056, A090057, A090058, A090059, A090060.

A090061 Numbers k divisible by exactly two nontrivial permutations (rearrangements) of the digits of k, excluding all permutations that result in digit loss.

Original entry on oeis.org

571428, 867132, 874125, 923076, 5179428, 5714028, 5714280, 5714820, 5719428, 5971428, 8524710, 8571042, 8671320, 8679132, 8741250, 8749125, 8914752, 8957142, 9230760, 9239076, 37451268, 41957028, 42195708, 42713568, 42915780, 42971580, 43157286, 43751286, 48713562, 51374268

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial permutations are identified as those where the permutation = k itself. Digit loss occurs when a permutation has 0 in the most significant position, which drops off, leaving a number with fewer digits. For example, when k is 3105, the permutation 0315 is excluded because 315 has fewer digits than 3105.

In the first million values of k, there is only one term that is divisible by three lossless nontrivial permutations. That term is 857142 which is divisible by 142857, 285714 and 428571. Note that 857142 is equal to floor((6/7)*10^6).

			a(4)=923076 is a term because 923076 is divisible by both 230769 and 307692, two nontrivial permutations of 923076 with the same number of digits.

David A. Corneth, Table of n, a(n) for n = 1..18986 (terms <= 10^10)
David A. Corneth, PARI program
Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090056, A090057.

PARI
```
\\ See Corneth link
```

a(5)-a(25) from Donovan Johnson, Sep 16 2009

More terms from David A. Corneth, Jun 08 2025

A090053 Numbers divisible by the number formed when their digits are sorted in ascending order, excluding trivial cases.

Original entry on oeis.org

105, 108, 405, 510, 540, 702, 703, 810, 1001, 1005, 1008, 1020, 1050, 1080, 2002, 2016, 2025, 2040, 2050, 2100, 3003, 3042, 3060, 3105, 3510, 4004, 4005, 4050, 4080, 4200, 5005, 5010, 5040, 5049, 5100, 5130, 5200, 5400, 6006, 6084

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial cases are identified as (1) values of k where the digits are already in ascending order, like 123 or 2228, such that ASort(k)=k, or (2) values of k where k mod 10 = 0 and all digits other than trailing zeros are in ascending order, like 12000 or 333500, such that ASort(k)=k/10^z, where z = the number of trailing zeros of k. In case (1), k/ASort(k) is equivalent to k/k (as in 123/123). In case (2), k/ASort(k) is 10^z (as in 12000/12). Neither of these cases is very interesting.

Sequence A084687 is a subsequence of this sequence, but that sequence excludes any value of k with 1 or more zero digits.

			a(1)=105 because the digits of 105 in ascending order are 015 and 105 is divisible by 15. a(24)=3105 because the digits of 3105 in ascending order are 135 and 3105 is divisible by 135.

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A084687, A090055, A090056.

A382946 a(n) is the least positive integer k having a proper divisor d such that the base n expansions of k and d, without leading zeros, have, up to order, the same digits, or a(n) = -1 if no such k exists.

Original entry on oeis.org

-1, 64, 36, 16, 700, 36, 42, 64, 3105, 45, 594, 105, 130, 168, 945, 120, 1666, 96, 266, 275, 2457, 231, 460, 351, 450, 273, 7938, 175, 7714, 280, 682, 1024, 308, 459, 7525, 741, 962, 665, 27300, 288, 17097, 560, 1290, 1265, 18540, 1035, 1974, 540, 952, 715

Offset: 2

Rémy Sigrist, Apr 09 2025

Conjecture: a(n) > 0 for any n > 2.

			The first terms, alongside an appropriate divisor d, in bases 10 and n, are:
  n   a(n)  d     n in base n  d in base n
  --  ----  ----  -----------  -----------
   2    -1  N/A   N/A          N/A
   3    64    32  2,1,0,1      1,0,1,2
   4    36    18  2,1,0        1,0,2
   5    16     8  3,1          1,3
   6   700   350  3,1,2,4      1,3,4,2
   7    36    12  5,1          1,5
   8    42    21  5,2          2,5
   9    64    16  7,1          1,7
  10  3105  1035  3,1,0,5      1,0,3,5
  11    45    15  4,1          1,4
  12   594   198  4,1,6        1,4,6
  13   105    21  8,1          1,8
  14   130    65  9,4          4,9
  15   168    56  11,3         3,11
  16   945   315  3,11,1       1,3,11

Rémy Sigrist, Table of n, a(n) for n = 2..1000

Cf. A023094, A090056, A382945.

a(n) = {
    if (n==2, return (-1));
    for (k = 1, oo,
        my (t = vecsort(digits(k, n)));
        fordiv (k, d,
            if (d < k && vecsort(digits(d, n))==t,
                return (k);););); }

from sympy import divisors
from sympy.ntheory import digits
from itertools import count
def a(n):
    if n == 2:
        return -1
    for k in count(2*n):
        divs, kdigs = divisors(k), sorted(digits(k, n)[1:])
        for d in sorted(divs[:-1], reverse=True):
            ddigs = sorted(digits(d, n)[1:])
            if ddigs == kdigs:
                return k
            if len(ddigs) < len(kdigs):
                break
print([a(n) for n in range(2, 52)]) # Michael S. Branicky, Apr 13 2025

cp's OEIS Frontend

A090055 Numbers n divisible by at least one nontrivial permutation (rearrangement) of the digits of n.

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A090061 Numbers k divisible by exactly two nontrivial permutations (rearrangements) of the digits of k, excluding all permutations that result in digit loss.

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A090053 Numbers divisible by the number formed when their digits are sorted in ascending order, excluding trivial cases.

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A382946 a(n) is the least positive integer k having a proper divisor d such that the base n expansions of k and d, without leading zeros, have, up to order, the same digits, or a(n) = -1 if no such k exists.

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