A090061 -id:A090061 - cp's OEIS frontend

A090056 Numbers k divisible by at least one nontrivial permutation (rearrangement) of the digits of k, excluding all permutations that result in digit loss.

3105, 7128, 7425, 8316, 8712, 9513, 9801, 30105, 31050, 37125, 42741, 44172, 67128, 70416, 71208, 71253, 71280, 71328, 71928, 72108, 72441, 74142, 74250, 74628, 74925, 78912, 79128, 80712, 81816, 82755, 83160, 83181, 83916, 84510, 85725, 86712, 87120, 87132, 87192, 87912

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. These exclusions make this sequence a subsequence of A090055. A084687 is a subsequence of this sequence.
Apparently each term of this sequence is divisible by 3. This has been confirmed for the first 100 terms.
From David A. Corneth, Jun 08 2025: (Start)
All terms are divisible by 3. Proof: Suppose a term t is not divisible by 3.
Then for some m < t that is an anagram of t with the same number of digits as t we have m * c = t where 2 <= c <= 9. If c > 9 then t has more digits than m and if c = 1 then m is a trivial anagram of t, excluded by definition.
Since m and t are anagrams, 9 | (t - m) = (c - 1)*m. If t is not divisible by 3 then m is not divisible by 3 and so 9 | c - 1. This is a contradiction since 2 <= c <= 9 for which no c is divisible by 9 which completes the proof.
In addition if t is not divisible by 9 then c = 4 or 7. (End)

			a(1)=3105 because 3105 is divisible by 1035, a nontrivial permutation of 3105 with the same number of digits.
a(4)=8316 because 8316 is divisible by 1386, a nontrivial permutation of 8316 with the same number of digits.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 362 terms from Harvey P. Dale)
David A. Corneth, PARI program
C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A084687, A090055, A090061, A179239.

dnpQ[n_]:=Module[{d=FromDigits/@Select[Permutations[IntegerDigits[n]], First[#]>0&&Reverse[#]!=#&]},Count[Divisible[n,d],True]>1]; Select[ Range[90000],dnpQ] (* Harvey P. Dale, Aug 19 2013 *)

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

More terms from David A. Corneth, Jun 08 2025

A090057 Numbers n divisible by exactly two nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

1050, 1080, 3105, 5100, 5400, 7020, 7030, 9207, 9801, 10010, 10050, 10080, 10098, 10200, 10206, 20020, 20160, 20250, 20304, 20400, 20500, 20790, 21000, 21060, 30015, 30030, 30105, 30240, 30420, 30450, 30600, 35100, 40040, 40050

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.

			a(3)=3105 because 3105 is divisible by both 135 and 1035, two nontrivial permutations of 3105. a(8)=9207 because 9207 is divisible by both 279 and 297, two nontrivial permutations of 9207.

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090058, A090059, A090060, A090061.

A090058 Numbers n divisible by exactly three nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

4050, 8100, 10500, 10530, 10800, 34020, 51000, 54000, 60912, 68040, 70300, 80190, 95040, 100100, 100500, 100800, 102000, 105030

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.

			60912 is a term because 60912 is divisible by 1269, 1296 and 1692, three nontrivial permutations of 60912.

Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090057, A090059, A090060, A090061.

A090059 Numbers n divisible by exactly four nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

31050, 50490, 70200, 92070, 100035, 100980, 102060, 105000, 108000

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.

			50490 is a term because 50490 is divisible by 459, 495, 594 and 4590, four nontrivial permutations of 50490.

Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090057, A090058, A090060, A090061.

A090060 Numbers n divisible by exactly five nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

40500, 81000, 98010, 105300

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.

			98010 is a term because 98010 is divisible by 198, 891, 1089, 8910 and 10890, five nontrivial permutations of 98010.

Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090057, A090058, A090059, A090061.

A373407 Smallest positive integer k such that no more than n numbers (formed by multiplying k by a digit) are anagrams of k, or -1 if no such number exists.

Original entry on oeis.org

1, 1035, 123876, 1402857, 1037520684, 142857

Offset: 1

Jean-Marc Rebert, Jun 04 2024

For n = 2..6 all terms are divisible by 9.
For n >= 4, a(n) must be divisible by 9, or a(n) = -1, because all anagrams d*k of k for d = 2, 3, 5, 6, 8 and 9 are divisible by 9. Thus there are only 3 values of d, i.e., 1, 4 and 7, for which k*d must not be divisible by 9.
If a(n) exists for n > 1 then 9|a(n). Holds for n = 2 and n = 3 by inspection. Proof for n >= 4: if k*d is an anagram of k where 2 <= d <= 9 then k*d - k = k*(d-1) is a multiple of 9. For this to be true, k must be a multiple of 9 as d is not of the form 1 (mod 3) for all d. - David A. Corneth, Jun 04 2024
From Michael S. Branicky, Jun 07 2024: (Start)
The following were constructed from multiples of cyclic numbers (cf. A180340, Wikipedia):
a(6)  = 142857 = (10^6 - 1) / 7;
a(7) <= 1304347826086956521739 = 3*(10^22 - 1) / 23;
a(8) <= 1176470588235294 = 2*(10^16 - 1) / 17;
a(9) <= 105263157894736842 = 2*(10^18 - 1) / 19. (End)

			a(2) = 1035, because 1035 * 1 = 1035 and 1035 * 3 = 3105 are anagrams of 1035, and no other number 1035 * i with digit i is an anagram of 1035, and no lesser number verifies this property.
Table n, k, set of multipliers.
  1   1          [1]
  2   1035       [1, 3]
  3   123876     [1, 3, 7]
  4   1402857    [1, 2, 3, 5]
  5   1037520684 [1, 2, 4, 5, 8]
  6   142857     [1, 2, 3, 4, 5, 6]

Wikipedia, Cyclic numbers.

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023092, A023093.
Cf. A245680, A245682.
Cf. A090061.
Cf. A180340, A001913, A036275.

isok(k, n) = my(d=vecsort(digits(k))); sum(i=1, 9, vecsort(digits(k*i)) == d) == n; \\ Michel Marcus, Jun 04 2024

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A090056 Numbers k divisible by at least one nontrivial permutation (rearrangement) of the digits of k, excluding all permutations that result in digit loss.

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A090057 Numbers n divisible by exactly two nontrivial permutations (rearrangements) of the digits of n.

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A090058 Numbers n divisible by exactly three nontrivial permutations (rearrangements) of the digits of n.

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A090059 Numbers n divisible by exactly four nontrivial permutations (rearrangements) of the digits of n.

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A090060 Numbers n divisible by exactly five nontrivial permutations (rearrangements) of the digits of n.

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A373407 Smallest positive integer k such that no more than n numbers (formed by multiplying k by a digit) are anagrams of k, or -1 if no such number exists.

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