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A230040 Number of ways to write n = x + y + z with y <= z such that all the five numbers 6*x-1, 6*y-1, 6*z-1, 6*x*y-1 and 6*x*z-1 are Sophie Germain primes. 6

%I

%S 0,0,1,2,2,3,3,1,3,4,5,2,1,1,3,4,4,3,4,6,5,2,2,6,5,1,2,4,2,2,3,6,5,7,

%T 6,2,3,4,4,2,3,5,1,4,7,4,6,3,9,4,2,5,4,3,9,2,4,3,6,3,5,8,8,5,8,6,2,4,

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

%N Number of ways to write n = x + y + z with y <= z such that all the five numbers 6*x-1, 6*y-1, 6*z-1, 6*x*y-1 and 6*x*z-1 are Sophie Germain primes.

%C Conjecture: a(n) > 0 for all n > 2.

%C This implies that 6*n-3 with n > 2 can be expressed as a sum of three Sophie Germain primes (i.e., those primes p with 2*p+1 also prime).

%C We have verified the conjecture for n up to 10^8. Note that any Sophie Germain prime p > 3 has the form 6*k-1.

%H Zhi-Wei Sun, <a href="/A230040/b230040.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.

%e a(4) = 2, since 4 = 1 + 1 + 2 = 2 + 1 + 1, and 6*1-1=5 and 6*2-1=11 are Sophie Germain primes.

%e a(26) = 1, since 26 = 15 + 2 + 9, and all the five numbers 6*15-1=89, 6*2-1=11, 6*9-1=53, 6*15*2-1=179 and 6*15*9=809 are Sophie Germain primes.

%t SQ[n_]:=PrimeQ[n]&&PrimeQ[2n+1]

%t a[n_]:=Sum[If[SQ[6i-1]&&SQ[6j-1]&&SQ[6(n-i-j)-1]&&SQ[6i*j-1]&&SQ[6*i(n-i-j)-1],1,0],{i,1,n-2},{j,1,(n-i)/2}]

%t Table[a[n],{n,1,100}]

%Y Cf. A005384, A219842, A219864, A227938, A229969, A229974, A230037.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Oct 06 2013

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