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.

9, 28, 18, 16, 40, 36, 42, 64, 105, 45, 154, 105, 130, 168, 260, 120, 340, 96, 266, 275, 495, 231, 460, 351, 450, 273, 792, 175, 928, 280, 682, 1024, 308, 459, 1302, 741, 962, 665, 1612, 288, 1804, 560, 1290, 1265, 2139, 1035, 1974, 540, 952, 715, 2720, 585

Offset

2,1

Comments

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.

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 2..2501

Formula

a(n) <= n^3 + 1.

Example

The first terms, alongside an appropriate divisor d, in bases 10 and n, are:
  n   a(n)  d    a(n) in base n  d in base n
  --  ----  ---  --------------  -----------
   2     9    3  1,0,0,1         1,1
   3    28    4  1,0,0,1         1,1
   4    18    6  1,0,2           1,2
   5    16    8  3,1             1,3
   6    40   10  1,0,4           1,4
   7    36   12  5,1             1,5
   8    42   21  5,2             2,5
   9    64   16  7,1             1,7
  10   105   15  1,0,5           1,5
  11    45   15  4,1             1,4
  12   154   22  1,0,10          1,10
  13   105   21  8,1             1,8
  14   130   65  9,4             4,9
  15   168   56  11,3            3,11
  16   260   20  1,0,4           1,4

Prog

(PARI) a(n) = {
    for (k = 1, oo,
        my (t = vecsort(select(sign, digits(k, n))));
        fordiv (k, d,
            if ((k/d) != n^valuation(k/d, n)
                && vecsort(select(sign, digits(d, n)))==t,
                return (k); ); ); ); }

Crossrefs

Cf. A096092, A382946.

Sequence in context: A390665 A033479 A254139 * A034116 A031454 A044999

Adjacent sequences: A382942 A382943 A382944 * A382946 A382947 A382948

Keyword

nonn,base

Author

Rémy Sigrist, Apr 09 2025

Status

approved