cp's OEIS Frontend

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.

%I A227923 #32 Jul 07 2023 18:43:43
%S A227923 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,
%T A227923 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,
%U A227923 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
%N 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.
%C A227923 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.
%C A227923 (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).
%C A227923 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.
%C A227923 We have verified that a(n) > 0 for all n = 2..10^8.
%C A227923 Conjecture verified up to 10^9. - _Mauro Fiorentini_, Jul 07 2023
%H A227923 Zhi-Wei Sun, <a href="/A227923/b227923.txt">Table of n, a(n) for n = 1..10000</a>
%H A227923 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
%e A227923 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.
%e A227923 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.
%t A227923 SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[12n-1]
%t A227923 TQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
%t A227923 a[n_]:=Sum[If[SQ[i]&&TQ[n-i],1,0],{i,1,n-1}]
%t A227923 Table[a[n],{n,1,100}]
%Y A227923 Cf. A001359, A006512, A005384, A046132, A176130, A187757, A199920, A227920, A230037, A230040.
%K A227923 nonn
%O A227923 1,3
%A A227923 _Zhi-Wei Sun_, Oct 09 2013