A295483 Composite squarefree numbers k such that Sum_{i=1..j} (p_i)^k mod k = 0, where p_i is one of the j prime divisors of k.

290, 610, 1491, 24423, 55210, 738507, 3619317, 3668889, 384199202, 1307828445, 4664465273

Offset

1,1

Comments

Primes are excluded because they are a banal solution of the congruence.

Links

Table of n, a(n) for n=1..11.

Example

Prime factors of 290 are 2, 5, 29 and (2^290 + 5^290 + 29^290) mod 290 = 0.

Maple

with(numtheory): P:=proc(q) local a, k, n; for n from 2 to q do
if issqrfree(n) and not isprime(n) then a:=ifactors(n)[2];
if add(a[k][1]^n, k=1..nops(a)) mod n = 0 then print(n); fi; fi; od; end: P(10^6);

Mathematica

okQ[k_] := Module[{pp, ee}, {pp, ee} = FactorInteger[k] // Transpose; Max[ee] < 2 && Mod[Total[PowerMod[#, k, k]& /@ pp], k] == 0]; Reap[For[k = 6, k < 4*10^6, k++, If[CompositeQ[k], If[okQ[k], Print[k]; Sow[k] ] ] ] ][[2, 1]] (* Jean-François Alcover, Feb 15 2018 *)

Prog

(PARI) lista(nn) = {forcomposite(n=1, nn, if (issquarefree(n), f = factor(n); s = sum(k=1, #f~, Mod(f[k, 1], n)^n); if (lift(s) == 0, print1(n, ", ")); ); ); } \\ Michel Marcus, Feb 13 2018

Crossrefs

Subsequence of A120944.

Sequence in context: A296055 A090839 A158255 * A075299 A031712 A251049

Adjacent sequences: A295480 A295481 A295482 * A295484 A295485 A295486

Keyword

nonn,more

Author

Paolo P. Lava, Feb 13 2018

Extensions

a(6)-a(8) from Michel Marcus, Feb 13 2018

a(9)-a(11) from Giovanni Resta, Feb 13 2018

Status

approved