A379834 Positive integers with digits in nondescending order whose digit product is an integer power of their digit sum, given exponent > 1.

1, 999, 2558, 3366, 4444, 12489, 13377, 22444, 112668, 113388, 114455, 122289, 123336, 222244, 666666, 1113399, 1113468, 1114449, 1122455, 1133334, 1377789, 1555888, 2222224, 2346669, 3334689, 3344499, 4444448, 11112778, 11114466, 11122368, 11122449, 11222255, 11223333, 11467779, 12366666, 13336668

Offset

1,2

Comments

This sequence would include the terms of A274124 (digit product equal to digit sum) if the power were allowed to be 1.

This sequence is similar to A117720 (digit product is the square of the digit sum) except that powers > 2 are also included, and that A117720 does not require nondescending digits. These results are from empirical observation.

Each number of this sequence is an element of a group of permutations of its digits.

Since both addition and multiplication are commutative, it is only necessary to select one representative member of each digit permutation group. The member with digits in nondescending order is chosen.

Links

David A. Corneth, Table of n, a(n) for n = 1..10753 (first 1000 terms from Robert Israel, terms <= 10^37)

Robert Kinner, Pascal Program

Example

The product of the digits of 999 equals the sum of the digits, 27, squared.
The product of the digits of 666666 equals the cube of the sum of the digits, 36.

Maple

filter:= proc(n) local L, s, p;
  L:= convert(n, base, 10);
  s:= convert(L, `+`); p:= convert(L, `*`);
  p > s and p = s^padic:-ordp(p, s)
end proc:
R:= 1: count:= 1:
Cands:= [$1..9]:
for d from 2 to 8 do
  Cands:= map(proc(t) local i; seq(10*t+i, i=(t mod 10) .. 9) end proc, Cands);
  V:= select(filter, Cands);
  R:= R, op(V); count:= count+nops(V);
od:
R; # Robert Israel, May 15 2026

Prog

(Pascal) // See Robert Kinner link.

Crossrefs

Cf. A001414, A002473, A117720, A274124, A395064.

Sequence in context: A226752 A043527 A117720 * A110401 A395064 A213156

Adjacent sequences: A379831 A379832 A379833 * A379835 A379836 A379837

Keyword

nonn,base

Author

Robert Kinner, Jan 03 2025

Status

approved