%I #20 Mar 10 2015 01:53:51
%S 2,3,5,7,13,17,19,31,47,59,61,71,101,103,107,109,137,149,151,157,167,
%T 181,197,211,223,227,229,269,317,337,349,353,379,383,389,401,421,439,
%U 449,457,463,479,521,523,541,547,563,569,571,587,599,613,617,631,643
%N Primes p that divide A123855(p-1).
%C A123855(n) = Sum_{j=1..n} Sum_{i=1..n} prime(i)^j.
%C Prime p = a(n) divides A123855(p-1).
%C Nonprime numbers n that divide A123855(n-1) are listed in A123857.
%C It appears that 2^k divides A123855(2^k-1) for all k>0 (confirmed for 0<k<10).
%H M. F. Hasler, Nov 10 2006, <a href="/A123856/b123856.txt">Table of n, a(n) for n = 1..199</a>
%p A123855_mod := proc(n,p) option remember; local s,i,pi; s:=0: for i to n do pi:= ithprime(i) mod p: if pi=1 then s:=s+n mod p: else s := s+pi*(pi &^ n - 1)/(pi-1) mod p fi od end; A123856 := proc(n::posint) option remember; local p; if n>1 then p:=nextprime( procname(n-1)) else p:=2 fi: while A123855_mod(p-1,p)<>0 do p:=nextprime( p ) od: p end; # _M. F. Hasler_, Nov 10 2006
%t fQ[p_] := Mod[ Sum[ PowerMod[ Prime@ i, j, p], {j, p - 1}, {i, p - 1}], p] == 0; Select[ Prime@ Range@ 117, fQ] (* _Robert G. Wilson v_, Jun 10 2011 *)
%Y Cf. A123857, A123855, A086787.
%K nonn
%O 1,1
%A _Alexander Adamchuk_, Oct 13 2006