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A340541 Primes p such that the product of (p mod q) for primes q < p is a multiple of the sum of (p mod q) for primes q < p.

%I A340541 #13 Jan 12 2021 15:36:05
%S A340541 3,11,17,31,59,67,73,101,103,173,179,193,199,211,223,349,401,463,491,
%T A340541 499,557,563,569,571,577,587,607,613,619,631,673,709,751,757,769,797,
%U A340541 809,857,859,877,911,919,929,967,1009,1033,1039,1049,1151,1153,1193,1201,1237,1249,1259,1289,1297,1303
%N A340541 Primes p such that the product of (p mod q) for primes q < p is a multiple of the sum of (p mod q) for primes q < p.
%C A340541 Primes prime(n) such that A102647(n) is divisible by A033955(n).
%H A340541 Robert Israel, <a href="/A340541/b340541.txt">Table of n, a(n) for n = 1..1000</a>
%e A340541 a(3) = 17 is a term since (17 mod q) for primes q=2,3,5,7,11,13 are 1,2,2,3,6,4, and 1*2*2*3*6*4 = 288 is divisible by 1+2+2+3+6+4 = 18.
%p A340541 P:= [seq(ithprime(i),i=1..1000)]:
%p A340541 filter:= proc(n) local L,k;
%p A340541   L:= [seq(P[n] mod P[k],k=1..n-1)];
%p A340541   convert(L,`*`) mod convert(L,`+`) = 0
%p A340541 end proc:
%p A340541 S:=select(filter, [$2..1000]):
%p A340541 map(t -> P[t], S);
%o A340541 (PARI) isok(p) = {if (isprime(p) && (p>2), my(s=0, t=1); forprime(q=2, p-1, my(x= p%q); s += x; t *= x;); !(t % s););} \\ _Michel Marcus_, Jan 11 2021
%Y A340541 Cf. A033955, A102647.
%K A340541 nonn
%O A340541 1,1
%A A340541 _J. M. Bergot_ and _Robert Israel_, Jan 11 2021