A230507 Number of ways to write n = a + b + c with a <= b <= c, where a, b, c are among those numbers m (terms of A230506) with 2*m + 1 and 2*m^3 + 1 both prime.

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 2, 3, 3, 3, 3, 3, 2, 3, 3, 5, 4, 2, 2, 5, 5, 3, 3, 6, 7, 8, 4, 3, 7, 8, 6, 5, 6, 8, 9, 7, 4, 5, 8, 8, 7, 4, 5, 10, 9, 5, 4, 7, 8, 9, 6, 4, 8, 11, 7, 4, 5, 6, 10, 7, 2, 5, 8, 7, 5, 3, 3, 8, 8, 2, 3, 6, 4, 6, 3, 1, 5, 6, 3, 2, 3, 3, 7, 3, 1, 5, 5, 2, 4, 4, 4, 7, 5

Offset

1,8

Comments

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

(ii) Any integer n > 8 can be written as x + y + z (x, y, z > 0) with 2*x + 1, 2*y + 1, 2*z - 1, 2*x^4 - 1, 2*y^4 - 1, 2*z^4 - 1 all prime.

Either of the two parts of the conjecture is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.

Part (i) implies that there are infinitely many positive integers n with 2*n + 1 and 2*n^3 + 1 both prime, and part (ii) implies that there are infinitely many positive integers n with 2*n + 1 and 2*n^4 - 1 both prime.

We have verified the conjecture for n up to 10^6.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, On representations via sparse primes, a message to Number Theory List, Oct. 23, 2013.

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Example

a(8) = 2 since 8 = 1 + 1 + 6 = 1 + 2 + 5, and 2*1 + 1 = 3, 2*1^3 + 1 = 3, 2*6 + 1 = 13, 2*6^3 + 1 = 433, 2*2 + 1 = 5, 2*2^3 + 1 = 17, 2*5 + 1 = 11, 2*5^3 + 1 = 251 are all prime.

Mathematica

pp[n_]:=PrimeQ[2n+1]&&PrimeQ[2n^3+1]
a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j], 1, 0], {i, 1, n/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A068307, A230219, A230351, A230493, A230502, A230506.

Sequence in context: A328701 A085257 A352889 * A352888 A354267 A366804

Adjacent sequences: A230504 A230505 A230506 * A230508 A230509 A230510

Keyword

nonn

Author

Zhi-Wei Sun, Oct 21 2013

Status

approved