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.

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, 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, 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

Offset: 1

Author

J. M. Bergot and Robert Israel, Dec 28 2021

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for n >= 3.

a(n) <= A002375(n) with equality for n in A350398.

Examples

			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).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002375, A350398, A350400.

Programs

Maple

f:= proc(k) local P,i;
  P:= select(t -> isprime(t) and isprime(2*k-t) and isprime(-t^2 mod (2*k)), [2,seq(i,i=3..k,2)]);
  nops(P);
end proc:
map(f, [$1..100]);

Mathematica

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 *)