A237815 Number of primes p < n such that the number of Sophie Germain primes among 1, ..., n-p is a Sophie Germain prime.

0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 5, 5, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 3, 3, 4, 4, 3, 3, 3

Offset: 1

Author

Zhi-Wei Sun, Feb 13 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 4.

(ii) For any integer n > 11, there is a prime p < n such that the number of Sophie Germain primes among 1, ..., n-p is a square.

See also A237817 for a similar conjecture involving twin primes.

Examples

			a(5) = 1 since there are exactly two Sophie Germain primes not exceeding 5-2 = 3, and 2 is a Sophie Germain prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A005384, A000290, A237705, A237706, A237768, A237817.

Programs

Mathematica

sg[n_]:=PrimeQ[n]&&PrimeQ[2n+1]
sum[n_]:=Sum[If[PrimeQ[2Prime[k]+1],1,0],{k,1,PrimePi[n]}]
a[n_]:=Sum[If[sg[sum[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]