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

Author

Chuck Seggelin, Nov 21 2003

Keywords

• nonn
• base

Comments

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)

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A084687, A090055, A090061, A179239.

Programs

Mathematica

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

Extensions

More terms from David A. Corneth, Jun 08 2025