A382946 a(n) is the least positive integer k having a proper divisor d such that the base n expansions of k and d, without leading zeros, have, up to order, the same digits, or a(n) = -1 if no such k exists.

-1, 64, 36, 16, 700, 36, 42, 64, 3105, 45, 594, 105, 130, 168, 945, 120, 1666, 96, 266, 275, 2457, 231, 460, 351, 450, 273, 7938, 175, 7714, 280, 682, 1024, 308, 459, 7525, 741, 962, 665, 27300, 288, 17097, 560, 1290, 1265, 18540, 1035, 1974, 540, 952, 715

Offset

2,2

Comments

Conjecture: a(n) > 0 for any n > 2.

Links

Rémy Sigrist, Table of n, a(n) for n = 2..1000

Example

The first terms, alongside an appropriate divisor d, in bases 10 and n, are:
  n   a(n)  d     n in base n  d in base n
  --  ----  ----  -----------  -----------
   2    -1  N/A   N/A          N/A
   3    64    32  2,1,0,1      1,0,1,2
   4    36    18  2,1,0        1,0,2
   5    16     8  3,1          1,3
   6   700   350  3,1,2,4      1,3,4,2
   7    36    12  5,1          1,5
   8    42    21  5,2          2,5
   9    64    16  7,1          1,7
  10  3105  1035  3,1,0,5      1,0,3,5
  11    45    15  4,1          1,4
  12   594   198  4,1,6        1,4,6
  13   105    21  8,1          1,8
  14   130    65  9,4          4,9
  15   168    56  11,3         3,11
  16   945   315  3,11,1       1,3,11

Prog

(PARI) a(n) = {
    if (n==2, return (-1));
    for (k = 1, oo,
        my (t = vecsort(digits(k, n)));
        fordiv (k, d,
            if (d < k && vecsort(digits(d, n))==t,
                return (k); ); ); ); }
(Python)
from sympy import divisors
from sympy.ntheory import digits
from itertools import count
def a(n):
    if n == 2:
        return -1
    for k in count(2*n):
        divs, kdigs = divisors(k), sorted(digits(k, n)[1:])
        for d in sorted(divs[:-1], reverse=True):
            ddigs = sorted(digits(d, n)[1:])
            if ddigs == kdigs:
                return k
            if len(ddigs) < len(kdigs):
                break
print([a(n) for n in range(2, 52)]) # Michael S. Branicky, Apr 13 2025

Crossrefs

Cf. A023094, A090056, A382945.

Sequence in context: A073327 A169639 A252486 * A188828 A331224 A123994

Adjacent sequences: A382943 A382944 A382945 * A382947 A382948 A382949

Keyword

sign,base

Author

Rémy Sigrist, Apr 09 2025

Status

approved