A371981 Number of primes between two successive Sophie Germain primes, with Sophie Germain primes not themselves included in the count.

0, 0, 1, 3, 0, 2, 2, 6, 0, 5, 1, 7, 0, 1, 7, 0, 1, 5, 1, 9, 8, 1, 2, 7, 2, 10, 7, 2, 0, 3, 3, 3, 2, 4, 15, 5, 7, 0, 1, 2, 8, 14, 0, 7, 13, 4, 1, 3, 4, 0, 5, 3, 1, 17, 9, 9, 0, 2, 3, 5, 4, 1, 0, 7, 2, 14, 7, 2, 6, 0, 6, 7, 0, 18, 0, 6, 1, 7, 9, 3, 2, 0, 5, 28, 5, 3, 3, 2, 1, 5, 6, 7, 3, 15, 2

Offset: 1

Author

Alexandre Herrera, Apr 15 2024

Keywords

• nonn

Comments

Number of primes between A005384(n) and A005384(n+1).

Examples

			a(4) = 3 because there are 3 primes between 11 and 23: 13, 17 and 19.
		

Crossrefs

Cf. A000720, A005384.

Programs

Mathematica

-1 + Subtract @@ Map[PrimePi, {Last[#], First[#]}] & /@ Partition[Select[Prime[Range[500]], PrimeQ[2 # + 1] &], 2, 1] (* Michael De Vlieger, Apr 19 2024 *)

PARI

lista(nn) = my(vp = select(p->isprime(2*p+1), primes(nn)), wp = apply(primepi, vp)); vector(#wp-1, k, wp[k+1]-wp[k]-1); \\ Michel Marcus, May 21 2024

Python

from sympy import isprime
l = []
s = 0
for i in range(3,3800):
    if isprime(i):
        if isprime(2*i + 1):
            l.append(s)
            s = 0
        else:
            s += 1
print(l)

Formula

a(n) = A000720(A005384(n+1)) - A000720(A005384(n)) - 1. - Michael De Vlieger, Apr 19 2024