A305502 Number of ways to write n as x + y with 0 < x <= y such that x^3 + n*y^2 is prime.

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 1, 4, 2, 2, 1, 4, 1, 6, 2, 5, 2, 7, 1, 3, 2, 6, 2, 7, 2, 2, 4, 3, 3, 3, 3, 6, 1, 7, 3, 6, 7, 6, 3, 4, 1, 6, 2, 8, 3, 5, 3, 7, 5, 8, 4, 5, 3, 7, 4, 6, 4, 7, 5, 7, 5, 11, 6, 8, 4, 9, 3, 8, 4, 6, 5, 9, 5, 11, 6, 6, 6, 16, 7, 10, 5

Offset: 1

Author

Zhi-Wei Sun, Jun 03 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 20, 22, 28, 42, 50.

We have verified a(n) > 0 for all n = 2..10^7.

See also A232174 for a similar conjecture.

Examples

			a(2) = 1 since 2 = 1 + 1 with 1^3 + 2*1^2 = 3 prime.
a(3) = 1 since 3 = 1 + 2 with 1^3 + 3*2^2 = 13 prime.
a(11) = 1 since 11 = 5 + 6 with 5^3 + 11*6^2 = 521 prime.
a(20) = 1 since 20 = 3 + 17 with 3^3 + 20*17^2 = 5807 prime.
a(28) = 1 since 28 = 9 + 19 with 9^3 + 28*19^2 = 10837 prime.
a(42) = 1 since 42 = 19 + 23 with 19^3 + 42*23^2 = 29077 prime.
a(50) = 1 since 50 = 3 + 47 with 3^3 + 50*47^2 = 110477 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A000040, A000290, A000578, A232174.

Programs

Mathematica

tab={};Do[r=0;Do[If[PrimeQ[x^3+n(n-x)^2],r=r+1],{x,1,n/2}];tab=Append[tab,r],{n,1,90}];Print[tab]