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A382945 a(n) is the least positive integer k having a divisor d such that k/d is not a power of n and the base n expansions of k and d, possibly with leading zeros, have, up to order, the same digits.

%I A382945 #13 Apr 14 2025 09:07:52
%S A382945 9,28,18,16,40,36,42,64,105,45,154,105,130,168,260,120,340,96,266,275,
%T A382945 495,231,460,351,450,273,792,175,928,280,682,1024,308,459,1302,741,
%U A382945 962,665,1612,288,1804,560,1290,1265,2139,1035,1974,540,952,715,2720,585
%N A382945 a(n) is the least positive integer k having a divisor d such that k/d is not a power of n and the base n expansions of k and d, possibly with leading zeros, have, up to order, the same digits.
%C A382945 If k/d is a power of n, then k = d * n^z for some z >= 0 and, trivially, the base n expansions of k (with z trailing zeros) and d (with z leading zeros) have the same digits.
%C A382945 The sequence is well defined: for any n > 1, n^3 + 1 is divisible by n + 1, (n^3 + 1) / (n + 1) is not a power of n, and the base n expansions of n^3 + 1 and n + 1 have, up to order, the same digits.
%H A382945 Rémy Sigrist, <a href="/A382945/b382945.txt">Table of n, a(n) for n = 2..2501</a>
%F A382945 a(n) <= n^3 + 1.
%e A382945 The first terms, alongside an appropriate divisor d, in bases 10 and n, are:
%e A382945   n   a(n)  d    a(n) in base n  d in base n
%e A382945   --  ----  ---  --------------  -----------
%e A382945    2     9    3  1,0,0,1         1,1
%e A382945    3    28    4  1,0,0,1         1,1
%e A382945    4    18    6  1,0,2           1,2
%e A382945    5    16    8  3,1             1,3
%e A382945    6    40   10  1,0,4           1,4
%e A382945    7    36   12  5,1             1,5
%e A382945    8    42   21  5,2             2,5
%e A382945    9    64   16  7,1             1,7
%e A382945   10   105   15  1,0,5           1,5
%e A382945   11    45   15  4,1             1,4
%e A382945   12   154   22  1,0,10          1,10
%e A382945   13   105   21  8,1             1,8
%e A382945   14   130   65  9,4             4,9
%e A382945   15   168   56  11,3            3,11
%e A382945   16   260   20  1,0,4           1,4
%o A382945 (PARI) a(n) = {
%o A382945     for (k = 1, oo,
%o A382945         my (t = vecsort(select(sign, digits(k, n))));
%o A382945         fordiv (k, d,
%o A382945             if ((k/d) != n^valuation(k/d, n)
%o A382945                 && vecsort(select(sign, digits(d, n)))==t,
%o A382945                 return (k);););); }
%Y A382945 Cf. A096092, A382946.
%K A382945 nonn,base
%O A382945 2,1
%A A382945 _Rémy Sigrist_, Apr 09 2025