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%I #19 Feb 15 2018 08:13:46
%S 290,610,1491,24423,55210,738507,3619317,3668889,384199202,1307828445,
%T 4664465273
%N 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.
%C Primes are excluded because they are a banal solution of the congruence.
%e Prime factors of 290 are 2, 5, 29 and (2^290 + 5^290 + 29^290) mod 290 = 0.
%p with(numtheory): P:=proc(q) local a,k,n; for n from 2 to q do
%p if issqrfree(n) and not isprime(n) then a:=ifactors(n)[2];
%p if add(a[k][1]^n,k=1..nops(a)) mod n = 0 then print(n); fi; fi; od; end: P(10^6);
%t 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 *)
%o (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
%Y Subsequence of A120944.
%K nonn,more
%O 1,1
%A _Paolo P. Lava_, Feb 13 2018
%E a(6)-a(8) from _Michel Marcus_, Feb 13 2018
%E a(9)-a(11) from _Giovanni Resta_, Feb 13 2018
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