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.

%I A090056 #21 Jun 13 2025 16:37:32
%S A090056 3105,7128,7425,8316,8712,9513,9801,30105,31050,37125,42741,44172,
%T A090056 67128,70416,71208,71253,71280,71328,71928,72108,72441,74142,74250,
%U A090056 74628,74925,78912,79128,80712,81816,82755,83160,83181,83916,84510,85725,86712,87120,87132,87192,87912
%N A090056 Numbers k divisible by at least one nontrivial permutation (rearrangement) of the digits of k, excluding all permutations that result in digit loss.
%C A090056 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.
%C A090056 Apparently each term of this sequence is divisible by 3. This has been confirmed for the first 100 terms.
%C A090056 From _David A. Corneth_, Jun 08 2025: (Start)
%C A090056 All terms are divisible by 3. Proof: Suppose a term t is not divisible by 3.
%C A090056 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.
%C A090056 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.
%C A090056 In addition if t is not divisible by 9 then c = 4 or 7. (End)
%H A090056 David A. Corneth, <a href="/A090056/b090056.txt">Table of n, a(n) for n = 1..10000</a> (first 362 terms from Harvey P. Dale)
%H A090056 David A. Corneth, <a href="/A090056/a090056.gp.txt">PARI program</a>
%H A090056 C. Seggelin, <a href="http://www.plastereddragon.com/maths/asortdiv.htm">Numbers Divisible by Digit Permutations</a>.
%e A090056 a(1)=3105 because 3105 is divisible by 1035, a nontrivial permutation of 3105 with the same number of digits.
%e A090056 a(4)=8316 because 8316 is divisible by 1386, a nontrivial permutation of 8316 with the same number of digits.
%t A090056 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 *)
%o A090056 (PARI) \\ See Corneth link
%Y A090056 Cf. A084687, A090055, A090061, A179239.
%K A090056 nonn,base
%O A090056 1,1
%A A090056 _Chuck Seggelin_, Nov 21 2003
%E A090056 More terms from _David A. Corneth_, Jun 08 2025