A290399 - OEIS

%I #15 Oct 21 2022 19:46:43

%S 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,

%T 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,

%U 1,8,6,5,4,5,4,7,4,9,6,6,5,8,5,7,6,4

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

%H Tianxin Cai and Deyi Chen, <a href="https://doi.org/10.1090/S0025-5718-2013-02685-3">A new variant of the Hilbert-Waring problem</a>, Math. Comp. 82 (2013), 2333-2341.

%e 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.

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

%Y Cf. A000040, A000578, A233386.

%K nonn

%O 1,11

%A _XU Pingya_, Jul 29 2017