A354139 a(n) is the least positive integer m such that (k+1)^n + (k+2)^n + ... + (k+m)^n == 0 (mod n) for every positive integer k.

1, 4, 3, 8, 5, 36, 7, 16, 3, 20, 11, 72, 13, 28, 15, 32, 17, 108, 19, 200, 21, 44, 23, 144, 5, 52, 3, 56, 29, 180, 31, 64, 33, 68, 35, 216, 37, 76, 39, 400, 41, 1764, 43, 88, 15, 92, 47, 288, 7, 20, 51, 104, 53, 324, 55, 112, 57, 116, 59, 1800, 61, 124, 21, 128, 65, 396, 67, 136, 69, 140, 71

Offset

1,2

Comments

a(n) divides n * A007947(n).

Links

Table of n, a(n) for n=1..71.

Formula

a(2^t) = 2^(t+1) for integers t>0.

a(n) = A007947(n) for odd integers n.

Conjecture: a(n) = A007947(n) * A193267(n).

Example

a(2) = 4 because, for every positive integer k, (k+1)^2 + (k+2)^2 + (k+3)^2 + (k+4)^2 == 0 (mod 2), and no smaller positive integer satisfies this condition.

Mathematica

sum[n_, r_] := Mod[Sum[k^r, {k, 1, n}], r];
rad[r_] := Product[i[[1]], {i, FactorInteger[r]}];
seq[r_] := Table[sum[n, r], {n, 1, r*rad[r]}];
A354139[r_] := Piecewise[   {    {rad[r], OddQ[r]},
    {2*r, EvenQ[r] && PrimePowerQ[r]},
    {Length[FindRepeat[seq[r]]], EvenQ[r] && Not[PrimePowerQ[r]]}
    }
   ];
Table[A354139[r], {r, 1, 20}] (* Improved by Dimitrios T. Tambakos, Feb 08 2023 *)

Prog

(PARI) isok(k, n) = my(p=sum(i=1, k, Mod(i+x, n)^n)); if (p==0, return(1)); for (i=1, n, if (subst(p, x, i) != 0, return(0))); return(1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 21 2022

Crossrefs

Sequence in context: A082895 A086938 A007015 * A114562 A189042 A011451

Adjacent sequences: A354136 A354137 A354138 * A354140 A354141 A354142

Keyword

nonn

Author

Dimitrios T. Tambakos, May 18 2022

Status

approved