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A350399 a(n) is the number of prime pairs (p,q) with p <= q, p+q = 2*n, and p*q mod (2*n) prime.

%I A350399 #14 Dec 30 2021 14:23:55
%S A350399 0,0,1,1,1,1,2,2,2,2,3,3,3,2,3,2,1,3,1,2,4,1,2,5,2,1,4,2,2,6,3,3,4,2,
%T A350399 4,6,2,3,5,3,4,8,3,1,9,2,3,6,3,4,5,2,4,6,4,4,8,5,2,7,3,3,10,1,2,6,2,2,
%U A350399 6,5,4,5,3,3,11,1,4,8,4,4,7,2,5,8,4,2,7,4,1,12,4,2,9,3,4,7,2,5
%N A350399 a(n) is the number of prime pairs (p,q) with p <= q, p+q = 2*n, and p*q mod (2*n) prime.
%C A350399 Conjecture: a(n) > 0 for n >= 3.
%C A350399 a(n) <= A002375(n) with equality for n in A350398.
%H A350399 Robert Israel, <a href="/A350399/b350399.txt">Table of n, a(n) for n = 1..10000</a>
%e A350399 a(7) = 2 because there are 2 such pairs, namely 14 = 3+13 = 7+7 with 3*13 == 5 (mod 14) and 7*7 == 7 (mod 14).
%p A350399 f:= proc(k) local P,i;
%p A350399   P:= select(t -> isprime(t) and isprime(2*k-t) and isprime(-t^2 mod (2*k)), [2,seq(i,i=3..k,2)]);
%p A350399   nops(P);
%p A350399 end proc:
%p A350399 map(f, [$1..100]);
%t A350399 a[n_] := Count[Select[Range[2, 2*n], PrimeQ], _?(# >= n && PrimeQ[2*n - #] && PrimeQ[Mod[#*(2*n - #), 2*n]] &)]; Array[a, 100] (* _Amiram Eldar_, Dec 28 2021 *)
%Y A350399 Cf. A002375, A350398,  A350400.
%K A350399 nonn
%O A350399 1,7
%A A350399 _J. M. Bergot_ and _Robert Israel_, Dec 28 2021