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A173897

a(n) is the number of Sophie Germain primes (A005384) between prime(n)^2 and prime(n+1)^2.

1, 2, 2, 4, 1, 7, 2, 5, 9, 2, 8, 9, 2, 10, 12, 12, 4, 16, 7, 6, 14, 11, 19, 16, 10, 6, 11, 9, 11, 49, 11, 18, 6, 43, 10, 21, 18, 15, 25, 21, 7, 43, 11, 19, 12, 53, 55, 18, 9, 20, 35, 9, 50, 31, 32, 28, 4, 38, 23, 15, 65, 74, 17, 12, 27, 90, 38, 63, 13, 29, 38, 51, 46, 39, 27, 38, 47, 28

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OFFSET

1,2

COMMENTS

If you graph a(n) versus n, an interesting pattern emerges. As you go farther along the n-axis, greater are the number of Sophie Germain primes, on average, within each interval obtained. The smallest count of 1 occurs twice: between squares of (2,3) and (11,13). I suspect the number of Sophie Germain primes within each interval will never be zero. If one could prove that there is at least 1 Sophie Germain prime within each interval, this would imply that Sophie Germain primes are infinite.

LINKS

J. S. Cheema, Table of n, a(n) for n = 1..10000

Wikipedia, Sophie Germain Primes

EXAMPLE

For n = 1, we consider the interval [2^2, 3^2], within which is one Sophie Germain prime, 5. Thus a(1) = 1.

PROG

(SageMath) A173897 = lambda n: len([p for p in prime_range(nth_prime(n)**2, nth_prime(n+1)**2) if is_prime(2*p+1)]) # D. S. McNeil, Dec 02 2010

(PARI) is_a005384(n) = ispseudoprime(2*n+1)

a(n) = my(i=0); forprime(q=prime(n)^2, prime(n+1)^2, if(is_a005384(q) && q < prime(n+1)^2, i++)); i \\ Felix Fröhlich, Sep 04 2016

CROSSREFS

Cf. A005384.

Cf. A069482 (prime(n+1)^2 - prime(n)^2). - Zak Seidov, Sep 04 2016