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

1,11

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A061357.

Sequence in context: A209777 A356931 A377129 * A145677 A128229 A132013

Adjacent sequences: A356861 A356862 A356863 * A356865 A356866 A356867

Keyword

nonn

Author

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

Status

approved