OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0. Also, any odd integer greater than one can be written as x + y (0 < x < y) with x^3 + y^2 prime.
The conjecture implies that there are infinitely many primes of the form x^3 + y^2 (x, y > 0) with x^2 + y^2 also prime.
Note that Ming-Zhi Zhang ever asked (cf. A036468) whether any odd integer greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.
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(10) = 1 since 2*10 + 1 = 1 + 20 with 1^2 + 20^2 = 1^3 + 20^2 = 401 prime.
a(15) = 1 since 2*15 + 1 = 25 + 6 with 25^2 + 6^2 = 661 and 25^3 + 6^2 = 15661 both prime.
a(40) = 1 since 2*40 + 1 = 55 + 26 with 55^2 + 26^2 = 3701 and 55^3 + 26^2 = 167051 both prime.
a(91) =1 since 2*91 + 1 = 85 + 98 with 85^2 + 98^2 = 16829 and 85^3 + 98^2 = 623729 both prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[x^3+(2n+1-x)^2]&&PrimeQ[x^2+(2n+1-x)^2], 1, 0], {x, 1, 2n}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 22 2013
STATUS
approved