A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 3, 5, 5, 5, 3, 3, 5, 4, 4, 5, 6, 7, 4, 4, 5, 2, 6, 5, 5, 5, 4, 2, 4, 6, 4, 5, 4, 4, 8, 6, 5, 11, 6, 6, 8, 10, 5, 5, 5, 8, 6, 6, 11, 7, 5, 7, 9, 7, 6, 7, 8, 9, 6, 8, 10, 7, 11, 8, 7, 10, 9, 9, 6, 5, 7, 8, 13, 7, 9, 13, 13, 12, 9, 9

Offset

1,6

Comments

Conjecture: a(n)>0 for every n=1,2,3,... Moreover, any integer n>3 not among 7, 22, 31 can be written as p+q (q>0) with p and p^3+2*q^3 both prime.

We have verified this conjecture for n up to 10^8. D. R. Heath-Brown proved in 2001 that there are infinitely many primes in the form x^3+2*y^3, where x and y are positive integers.

Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y (x>=0, y>=0) with x^m+2*y^m prime.

When m=1, this follows from Bertrand's postulate proved by Chebyshev in 1850. For m = 5, 7, 9, 11, 13, 15, 17, 19, it suffices to require that n is greater than 46, 69, 141, 274, 243, 189, 320, 454 respectively.

Links

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

D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.

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

Example

a(9)=1 since 9=7+2 with 7^3+2*2^3=359 prime.
a(22)=1 since 22=1+21 with 1^3+2*21^3=18523 prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[k^3+2(n-k)^3]==True, 1, 0], {k, 0, n}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A220272, A219842, A219864, A219923.

Sequence in context: A112224 A058774 A033101 * A371128 A029217 A161230

Adjacent sequences: A220410 A220411 A220412 * A220414 A220415 A220416

Keyword

nonn

Author

Zhi-Wei Sun, Dec 13 2012

Status

approved