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