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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.

%I A354139 #43 Feb 18 2023 22:49:13
%S A354139 1,4,3,8,5,36,7,16,3,20,11,72,13,28,15,32,17,108,19,200,21,44,23,144,
%T A354139 5,52,3,56,29,180,31,64,33,68,35,216,37,76,39,400,41,1764,43,88,15,92,
%U A354139 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
%N 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.
%C A354139 a(n) divides n * A007947(n).
%F A354139 a(2^t) = 2^(t+1) for integers t>0.
%F A354139 a(n) = A007947(n) for odd integers n.
%F A354139 Conjecture: a(n) = A007947(n) * A193267(n).
%e A354139 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.
%t A354139 sum[n_, r_] := Mod[Sum[k^r, {k, 1, n}], r];
%t A354139 rad[r_] := Product[i[[1]], {i, FactorInteger[r]}];
%t A354139 seq[r_] := Table[sum[n, r], {n, 1, r*rad[r]}];
%t A354139 A354139[r_] := Piecewise[   {    {rad[r], OddQ[r]},
%t A354139     {2*r, EvenQ[r] && PrimePowerQ[r]},
%t A354139     {Length[FindRepeat[seq[r]]], EvenQ[r] && Not[PrimePowerQ[r]]}
%t A354139     }
%t A354139    ];
%t A354139 Table[A354139[r], {r, 1, 20}] (* Improved by _Dimitrios T. Tambakos_, Feb 08 2023 *)
%o A354139 (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);
%o A354139 a(n) = my(k=1); while (!isok(k,n), k++); k; \\ _Michel Marcus_, May 21 2022
%K A354139 nonn
%O A354139 1,2
%A A354139 _Dimitrios T. Tambakos_, May 18 2022