A227923 Number of ways to write n = x + y (x, y > 0) such that 6*x-1 is a Sophie Germain prime and {6*y-1, 6*y+1} is a twin prime pair.

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 1, 4, 2, 4, 4, 2, 5, 3, 4, 4, 2, 5, 4, 4, 5, 1, 3, 3, 5, 8, 4, 7, 4, 3, 7, 2, 7, 6, 5, 8, 3, 6, 6, 4, 10, 4, 8, 5, 4, 10, 3, 9, 4, 4, 6, 1, 8, 5, 5, 8, 4, 4, 6, 3, 7, 1, 3, 5, 4, 10, 5, 7, 6, 3, 11, 3, 9, 5, 5, 6, 2, 7, 5, 5, 9, 4, 6, 4, 5, 9, 2, 6, 3, 4, 5, 2, 6, 7

Offset

1,3

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 4 not equal to 13 can be written as x + y with x and y distinct and greater than one such that 6*x-1 is a Sophie Germain prime and {6*y-1, 6*y+1} is a twin prime pair.

(ii) Any integer n > 1 can be written as x + y (x, y > 0) such that 6*x-1 is a Sophie Germain prime, and {6*y+1, 6*y+5} is a cousin prime pair (or {6*y-1, 6*y+5} is a sexy prime pair).

Part (i) of the conjecture implies that there are infinitely many Sophie Germain primes, and also infinitely many twin prime pairs. For example, if all twin primes does not exceed an integer N > 2, and (N+1)!/6 = x + y with 6*x-1 a Sophie Germain prime and {6*y-1, 6*y+1} a twin prime pair, then (N+1)! = (6*x-1) + (6*y+1) with 1 < 6*y+1 < N+1, hence we get a contradiction since (N+1)! - k is composite for every k = 2..N.

We have verified that a(n) > 0 for all n = 2..10^8.

Conjecture verified up to 10^9. - Mauro Fiorentini, Jul 07 2023

Links

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

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(5) = 2 since 5 = 2 + 3 = 4 + 1, and 6*2-1 = 11 and 6*4-1 = 23 are Sophie Germain primes, and {6*3-1, 6*3+1} = {17, 19} and {6*1-1, 6*1+1} = {5,7} are twin prime pairs.
a(28) = 1 since 28 = 5 + 23 with 6*5-1 = 29 a Sophie Germain prime and {6*23-1, 6*23+1} = {137, 139} a twin prime pair.

Mathematica

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[12n-1]
TQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
a[n_]:=Sum[If[SQ[i]&&TQ[n-i], 1, 0], {i, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A001359, A006512, A005384, A046132, A176130, A187757, A199920, A227920, A230037, A230040.

Sequence in context: A210568 A262887 A106432 * A346425 A029836 A004257

Adjacent sequences: A227920 A227921 A227922 * A227924 A227925 A227926

Keyword

nonn

Author

Zhi-Wei Sun, Oct 09 2013

Status

approved