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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 2*m + 1 and 2*m^3 + 1 both prime.
2
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, 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}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 21 2013
STATUS
approved