A046019 a(n) gives the number of different powers m^n for which the sum of the digits is equal to m > 0.

1, 9, 2, 6, 6, 5, 5, 9, 4, 4, 7, 4, 2, 12, 6, 8, 7, 5, 3, 10, 4, 4, 8, 4, 4, 14, 5, 3, 7, 6, 2, 11, 2, 8, 4, 6, 3, 9, 3, 3, 7, 2, 5, 10, 6, 4, 9, 9, 5, 12, 2, 4, 5, 5, 6, 3, 2, 7, 4, 5, 5, 6, 3, 4, 5, 5, 4, 9, 2, 6, 4, 3, 3, 6, 5, 6, 4, 4, 5, 9, 5, 3, 5, 5, 2, 6, 3, 7, 7, 4, 3, 8, 4, 4, 9, 6, 2, 8, 2, 5, 6, 3

Offset

0,2

Comments

Number of m >= 1 with m = sum of digits of m^n.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Formula

a(n) = 1 + A046471(n). - T. D. Noe, Nov 26 2008

Example

a(0) = 1 because sumdigits(m)^0 = 1 = m has only one solution, m = 1.
a(1) = 9 because sumdigits(m)^1 = m > 0 for the single digit numbers 1, ..., 9.
a(2) = 2 because sumdigits(m)^2 = m > 0 only for 1^2 = 1 and 9^2 = 81, 8 + 1 = 9.
a(7) = 9 because the only positive solutions for 7th powers are 1^7 = 1 and
  18^7 = 612220032 with 6 + 1 + 2 + 2 + 2 + 3 + 2 = 18,
  27^7 = 10460353203 with 1 + 4 + 6 + 3 + 5 + 3 + 2 + 3 = 27,
  31^7 = 27512614111 with 2 + 7 + 5 + 1 + 2 + 6 + 1 + 4 + 1 + 1 + 1 = 31,
  34^7 = 52523350144 with 5 + 2 + 5 + 2 + 3 + 3 + 5 + 1 + 4 + 4 = 34,
  43^7 = 271818611107 with 2 + 7 + 1 + 8 + 1 + 8 + 6 + 1 + 1 + 1 + 7 = 43,
  53^7 = 1174711139837 with 1 + 1 + 7 + 4 + 7 + 1 + 1 + 1 + 3 + 9 + 8 + 3 + 7 = 53,
  58^7 = 2207984167552 with 2 + 2 + 7 + 9 + 8 + 4 + 1 + 6 + 7 + 5 + 5 + 2 = 58, and
  68^7 = 6722988818432 with 6 + 7 + 2 + 2 + 9 + 8 + 8 + 8 + 1 + 8 + 4 + 3 + 2 = 68.
a(9) = 4 because the positive solutions for 9th powers are 1^9 = 1 and
  54^9 = 3904305912313344 with sumdigits(3904305912313344) = 54,
  71^9 = 45848500718449031 with sumdigits(45848500718449031) = 71, and
  81^9 = 150094635296999121 with sumdigits(150094635296999121) = 81.
See the table A152147 for other exponents.

Crossrefs

Cf. A046471, A124359, A152147 (table of m such that the sum of digits of m^n equals m).

Sequence in context: A239628 A107821 A154162 * A021523 A335589 A275647

Adjacent sequences: A046016 A046017 A046018 * A046020 A046021 A046022

Keyword

nonn,base

Author

David W. Wilson

Extensions

Examples provided by Paolo P. Lava, Oct 30 2006

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

Edited by M. F. Hasler, Jun 13 2026

Status

approved