A211395 Number of Sophie Germain primes between 2^n and 2^(n+1).

1, 1, 1, 1, 2, 2, 3, 7, 8, 13, 23, 41, 67, 111, 193, 360, 630, 1091, 1938, 3558, 6448, 11876, 21649, 40151, 73658, 135711, 251786, 468678, 875247, 1634069, 3060794, 5746245, 10806204, 20356921, 38433398, 72656139, 137562095, 260848098, 495343258, 941805467, 1792999074

Offset: 0

Author

Brad Clardy, Feb 08 2013

Keywords

• nonn

Comments

To be precise, the number of Sophie Germain primes p, 2^n < p <= 2^(n+1). Since 2 is a Sophie Germain prime, this precise definition is important only for determining a(0) and a(1). The alternative definition (with 2^n <= p < 2^(n+1)) would give the sequence 0, 2, 1, 1, 2, 2, 3, 7, 8, 13, 23, 41, 67, 111, 193, ...

The Sophie Germain primes p are in A005384. The corresponding primes s = 2p + 1 are called safe primes, and are in A005385. The number of safe primes between 2^(n+1) and 2^(n+2) is given by the sequence in the previous paragraph.

Links

• Paul D. Beale, A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, arXiv preprint arXiv:1411.2484 [physics.comp-ph], 2014-2015.
• Jetanat Datephanyawat and Paul D. Beale, Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers, arXiv:1811.11629 [cs.CR], 2018-2021.

Crossrefs

Cf. A005384, A005385, A211397.

Programs

Mathematica

nmax = 36; rtable = Table[0, {nmax}];
Do[r = 0;
  Do[If[And[PrimeQ[i], PrimeQ[2 i + 1]], r++], {i, 1 + 2^n,
    2^(n + 1)}]; Print[n, " ", r];
  rtable[[n + 1]] = r, {n, 0, nmax - 1}];
rtable (* Paul D. Beale, Sep 19 2014 *)

PARI

a211395(n) = {local(r,i); r=0; for(i=2^n+1, 2^(n+1), if(isprime(i)&&isprime(2*i+1), r=r+1)); r} \\ Michael B. Porter, Feb 08 2013

Formula

a(n) = A211397(n+1) - A211397(n). - Michel Marcus, Sep 22 2014

Extensions

a(29)-a(36) from Paul D. Beale, Sep 19 2014

a(37)-a(40) from Amiram Eldar, Jul 25 2025