A135740 Maximal sum of decimal digits of a^b for 1 <= a,b <= n.

1, 4, 9, 13, 13, 27, 36, 37, 45, 45, 62, 64, 71, 97, 99, 99, 109, 117, 127, 136, 136, 148, 153, 163, 171, 197, 197, 202, 224, 224, 236, 236, 251, 256, 281, 281, 302, 302, 306, 306, 315, 352, 352, 355, 360, 385, 385, 396, 406, 406, 431, 432, 437, 441, 469, 469

Offset

1,2

Comments

Clearly a(n) <= a(n+1). For what values of n do we have equality? Is there an explicit formula for a(n)?

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Project Euler, Problem 56: Considering natural numbers of the form a^b, find the maximum digital sum.

Example

a(3) = 9 = 2+7 is the digit sum of 3^2 and 3^3=27, all other a^b with a,b <= 3 have smaller digit sum.
a(4) = 13 = 2+5+6 is the digit sum of 4^4=256, all other a^b with a,b <= 4 have smaller digit sum.
a(5) = 13 since also for a,b <= 5 there is no higher digit sum (but the same is obtained for 5^4=625).

Maple

f:= proc(n) option remember;
  local a;
  max(procname(n-1), seq(convert(convert(a^n, base, 10), `+`), a=1..n), seq(convert(convert(n^a, base, 10), `+`),
    a=1..n));
end proc;
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jun 18 2025

Mathematica

a = {1}; For[n = 2, n < 100, n++, r = a[[ -1]]; For[j = 1, j < n + 1, j++, If[Max[Plus @@ IntegerDigits[n^j], Plus @@ IntegerDigits[j^n]] > r, r = Max[Plus @@ IntegerDigits[n^j], Plus @@ IntegerDigits[j^n]]]]; AppendTo[a, r]]; a (* Stefan Steinerberger, Dec 22 2007 *)

Prog

(PARI) digitsum(n, s)=n=[n]; while(n, n=divrem(n[1], 10); s+=n[2]); s
A135740(n)=vecmax(matrix(n, n, i, j, digitsum(i^j)))

Crossrefs

Sequence in context: A125848 A225752 A066588 * A312870 A312871 A247881

Adjacent sequences: A135737 A135738 A135739 * A135741 A135742 A135743

Keyword

base,nonn

Author

M. F. Hasler, Nov 30 2007

Extensions

More terms from Stefan Steinerberger, Dec 22 2007

Status

approved