A230506 -id:A230506 - cp's OEIS frontend

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 2m + 1 and 2m^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

Zhi-Wei Sun, Oct 21 2013

nonn

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.

			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.

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.

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

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}]

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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 2m + 1 and 2m^3 + 1 both prime.

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cp's OEIS Frontend

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.

Views

Author

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Mathematica

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 2m + 1 and 2m^3 + 1 both prime.