A354449 a(n) is the number of pairs of primes (p,q) with p<q such that p+q = 2*n and that 2*n+p, 2*n+q, p*q-2*n and p*q+2*n are primes.

0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset

1,15

Links

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

Example

a(15) = 2 as there are two such pairs, (7,23) and (13,17): 2*15+7 = 37, 2*15+23 = 53, 7*23-2*15 = 131, 7*23+2*15 = 191, 2*15+13 = 43, 2*15+17 = 47, 13*17-2*15 = 191 and 13*17+2*15 = 251 are all prime.

Maple

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

Mathematica

a[n_] := Sum[If[AllTrue[{k, 2*n - k, 2*n + k, 4*n - k, k*(2 n - k) - 2*n, k*(2 n - k) + 2*n}, PrimeQ], 1, 0], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, May 31 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, ispseudoprime(k) && ispseudoprime(2*n-k) && ispseudoprime(2*n+k) && ispseudoprime(4*n-k) && ispseudoprime(k*(2*n-k)-2*n) && ispseudoprime(k*(2*n-k)+2*n)) \\ adapted from Mathematica code, Felix Fröhlich, May 31 2022

Crossrefs

Cf. A045917, A354462.

Sequence in context: A349378 A331918 A089803 * A349436 A089811 A091888

Adjacent sequences: A354446 A354447 A354448 * A354450 A354451 A354452

Keyword

nonn

Author

J. M. Bergot and Robert Israel, May 31 2022

Status

approved