A356864 a(n) is the number of primes p < n such that 2*n-p and p*(2*n-p)+2*n are also prime.

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 3, 0, 2, 3, 0, 3, 4, 1, 1, 2, 1, 2, 3, 0, 0, 3, 1, 3, 1, 0, 5, 3, 0, 2, 1, 0, 3, 6, 0, 1, 2, 1, 1, 3, 0, 2, 2, 0, 2, 1, 1, 4, 6, 0, 2, 11, 0, 3, 3, 0, 2, 2, 0, 0, 2, 0, 4, 4, 0, 1, 3, 1, 5, 3, 0, 2, 8, 0, 2, 3, 0, 1, 5, 0, 0, 6, 1, 4, 5, 0, 3, 4, 0, 3, 1

Offset: 1

Author

J. M. Bergot and Robert Israel, Sep 01 2022

Keywords

• nonn

Comments

a(n) is the number of k such that n-k, n+k and n^2+2*n-k^2 are all prime.

If n == 1 (mod 3) then a(n) <= 1, as the only possible p is 3.

Examples

			a(11) = 2 because 3, 22-3 = 19 and 3*19+22 = 79, and 5, 22-5 = 17 and 5*17+22 = 107 are all prime.
		

Links

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

Crossrefs

Cf. A061357.

Programs

Maple

f:= proc(m) local p,q,t;
  p:= 1: t:= 0:
  do
    p:= nextprime(p);
    q:= n-p;
    if q <= p then return t fi;
    if isprime(q) and isprime(p*q+m) then t:= t+1 fi;
  od
end proc:
map(f, 2*[$1..100]);

Mathematica

a[n_] := Count[Range[n - 1], ?(AllTrue[{#, 2*n - #, #*(2*n - #) + 2*n}, PrimeQ] &)]; Array[a, 100] (* _Amiram Eldar, Sep 01 2022 *)