A237768 Number of primes p < n with pi(n-p) a Sophie Germain prime, where pi(.) is given by A000720.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 13 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 20, 21, 26, 27, 30, 31, 32, 60, 61, 68, 69, 80, 81.

This is stronger than part (i) of the conjecture in A237705.

We have verified that a(n) > 0 for all n = 5, ..., 2*10^7.

Examples

			a(5) = 1 since 2, pi(5-2) = pi(3) = 2 and 2*2 + 1 = 5 are all prime.
a(12) = 1 since 7, pi(12-7) = pi(5) = 3 and 2*3 + 1 = 7 are all prime.
a(81) = 1 since 47, pi(81-47) = pi(34) = 11 and 2*11 + 1 = 23 are all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000720, A005384, A237284, A237705, A237706.

Programs

Mathematica

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