A290399 Number of solutions to Diophantine equation x + y + z = prime(n) with x*y*z = k^3 (0 < x <= y <= z).

0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 3, 2, 1, 3, 2, 2, 2, 3, 1, 2, 3, 3, 3, 4, 3, 5, 2, 1, 5, 1, 4, 3, 3, 3, 3, 4, 5, 3, 3, 6, 3, 2, 3, 5, 5, 3, 6, 8, 2, 3, 7, 5, 7, 3, 5, 7, 5, 4, 1, 7, 4, 1, 8, 6, 5, 4, 5, 4, 7, 4, 9, 6, 6, 5, 8, 5, 7, 6, 4

Offset: 1

Author

XU Pingya, Jul 29 2017

Keywords

• nonn

Examples

			a(11) = 2 because the equation x + y + z = 31 (prime(11)) has exactly 2 solutions with x*y*z = k^3: (x, y, z) = (1, 5, 25) and (1, 12, 18), which satisfy 1*5*25 = 5^3 and 1*12*18 = 6^3.
		

Links

• Tianxin Cai and Deyi Chen, A new variant of the Hilbert-Waring problem, Math. Comp. 82 (2013), 2333-2341.

Crossrefs

Cf. A000040, A000578, A233386.

Programs

Mathematica

a[n_] := Length@ Select[ IntegerPartitions[ Prime[n], {3}], IntegerQ[ (Times @@ #)^(1/3)] &]; Array[a, 50] (* Giovanni Resta, Aug 07 2017 *)