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

1,6

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.

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

Example

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

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}]

Crossrefs

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

Sequence in context: A035486 A282347 A172397 * A238701 A238134 A143489

Adjacent sequences: A237812 A237813 A237814 * A237816 A237817 A237818

Keyword

nonn

Author

Zhi-Wei Sun, Feb 13 2014

Status

approved