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.

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, 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, 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

Offset

1,4

Comments

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

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).

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.

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.

Example

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.
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.

Mathematica

SQ[n_]:=PrimeQ[n]&&PrimeQ[2n+1]
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}]
Table[a[n], {n, 1, 100}]

Crossrefs

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

Sequence in context: A128924 A364569 A239957 * A242361 A116464 A346136

Adjacent sequences: A230037 A230038 A230039 * A230041 A230042 A230043

Keyword

nonn

Author

Zhi-Wei Sun, Oct 06 2013

Status

approved