A385314 a(n) is the least positive integer m such that Sum_{k = 1 .. m} k^n is divisible by n.

1, 3, 2, 7, 4, 4, 6, 15, 2, 4, 10, 8, 12, 3, 5, 31, 16, 27, 18, 24, 6, 11, 22, 31, 4, 12, 2, 7, 28, 4, 30, 63, 11, 8, 14, 40, 36, 19, 12, 31, 40, 8, 42, 16, 5, 11, 46, 31, 6, 4, 17, 32, 52, 40, 10, 31, 18, 28, 58, 31, 60, 15, 6, 127, 4, 27, 66, 8, 23, 7, 70, 80, 72, 36, 5, 47, 6, 36, 78, 31, 2

Offset: 1

Author

Robert Israel, Jun 25 2025

Keywords

• nonn

Comments

If p is an odd prime, a(p) = p - 1.

If n > 1 is odd, a(n) <= n - 1.

For all n, a(n) <= n^2 - 1.

Examples

			a(3) = 2 because 1^3 + 2^3 = 9 is divisible by 3, while 1^3 is not.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A057291, A057292, A103438.

Programs

Maple

f:= proc(n) local k,t;
  t:= 0:
  for k from 1 do
    t:= t + k &^ n mod n;
    if t = 0 then return k fi;
  od:
end proc:
map(f, [$1..100]);

Mathematica

a[n_]:=Module[{m=1},While[!Divisible[Sum[k^n,{k,1,m}],n],m++];m];Array[a,81] (* James C. McMahon, Jun 25 2025 *)

PARI

a(n) = my(m=1); while(sum(k=1, m, k^n) % n, m++); m; \\ Michel Marcus, Jun 25 2025