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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Oct 08 2013

Keywords

• nonn

Comments

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.

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.

Examples

			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.
		

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.

Crossrefs

Cf. A001359, A006512, A005384, A045536, A227923, A229969, A229974, A230037, A230040.

Programs

Mathematica

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