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

1,2

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.

Links

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A057291, A057292, A103438.

Sequence in context: A344969 A011772 A060451 * A220291 A129187 A166532

Adjacent sequences: A385311 A385312 A385313 * A385315 A385316 A385317

Keyword

nonn

Author

Robert Israel, Jun 25 2025

Status

approved