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

1,6

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.

Links

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A237721 A254296 A248371 * A237705 A340673 A031284

Adjacent sequences: A237765 A237766 A237767 * A237769 A237770 A237771

Keyword

nonn

Author

Zhi-Wei Sun, Feb 13 2014

Status

approved