A211395 - OEIS

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

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OFFSET

0,5

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

Table of n, a(n) for n=0..36.

Paul D. Beale, A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, arXiv preprint arXiv:1411.2484, 2014-2015.

Paul D. Beale, Jetanat Datephanyawat, Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers, arXiv:1811.11629 [cs.CR], 2018.

FORMULA

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

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}];