A237838 a(n) = |{0 < k <= n: the number of Sophie Germain primes among 1, ..., k*n is a Sophie Germain prime}|.

0, 1, 3, 2, 3, 2, 2, 2, 2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 2, 1, 3, 3, 4, 4, 4, 3, 4, 1, 1, 3, 3, 2, 3, 1, 1, 2, 2, 4, 1, 4, 3, 5, 4, 5, 4, 3, 4, 3, 4, 3, 2, 1, 4, 3, 4, 6, 1, 3, 3, 3, 4, 2, 3

Offset: 1

Author

Zhi-Wei Sun, Feb 14 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

See also A237839 for a similar conjecture involving twin primes.

Examples

			a(20) = 1 since 11 is a Sophie Germain prime, and there are exactly 11 Sophie Germain primes among 1, ..., 6*20 (namely, they are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005384, A237578, A237768, A237815, A237839.

Programs

Mathematica

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