cp's OEIS Frontend

A227920 Number of ways to write n = x + y + z with y and z distinct and greater than x such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.

%I A227920 #32 Apr 17 2016 09:21:11
%S A227920 0,0,0,0,0,1,1,3,1,3,1,2,4,1,3,1,3,4,1,4,2,5,4,1,4,4,3,5,1,3,2,3,8,2,
%T A227920 6,4,4,7,2,6,5,3,8,2,6,6,3,10,2,8,4,4,10,2,9,4,4,6,1,7,4,4,8,5,3,6,4,
%U A227920 7,1,3,5,2,10,3,7,5,3,11,3,9,4,5,6,1,7,5,5,9,4,6,4,6,9,2,5,4,3,5,2,6
%N A227920 Number of ways to write n = x + y + z with y and z distinct and greater than x such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.
%C A227920 By part (i) of the conjecture in the comments in A227923, for any integer n > 5 not equal to 14 we have a(n) > 0, because there are distinct positive integers x, y, z with x = 1 such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.
%C A227920 Conjecture: Any integer n > 2 can be written as x + y + z (x, y, z > 0) such that 6*x-1, 6*y-1, 6x*y-1, 6*z-1 are Sophie Germain primes, and {6*x-1, 6*x+1}, and {6*y-1, 6*y+1} are twin prime pairs.
%H A227920 Zhi-Wei Sun, <a href="/A227920/b227920.txt">Table of n, a(n) for n = 1..10000</a>
%H A227920 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e A227920 a(14) = 1 since 14 = 2 + 7 + 5, and 6*2-1 = 11, 6*7-1 = 41, 6*2*7-1 = 83 are Sophie Germain primes, and {6*2-1, 6*2+1} ={11, 13}, {6*5-1, 6*5+1} = {29, 31}, {6*2*5-1, 6*2*5+1} = {59, 61} are twin prime pairs.
%t A227920 SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[12n-1]
%t A227920 TQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
%t A227920 RQ[n_]:=TQ[n]&&PrimeQ[12n-1]
%t A227920 a[n_]:=Sum[If[RQ[i]&&SQ[j]&&SQ[i*j]&&TQ[n-i-j]&&TQ[i(n-i-j)]&&Abs[n-i-2j]>0,1,0],{i,1,n/3-1},{j,i+1,n-1-2i}]
%t A227920 Table[a[n],{n,1,100}]
%Y A227920 Cf. A001359, A006512, A005384, A045536, A227923, A229969, A229974, A230037, A230040.
%K A227920 nonn
%O A227920 1,8
%A A227920 _Zhi-Wei Sun_, Oct 08 2013