A395635 a(n) = number of triples (x, y, z) such that 2*x^2 + y*z = n, where x, y, z are positive integers with gcd(x,y*z)>1.

2, 0, 3, 0, 4, 0, 4, 0, 4, 0, 6, 2, 4, 0, 9, 0, 6, 3, 6, 0, 10, 0, 8, 4, 6, 0, 15, 0, 12, 4, 10, 0, 16, 0, 15, 4, 8, 0, 21, 0, 14, 4, 12, 0, 17, 2, 18, 4, 10, 0, 24, 0, 16, 6, 14, 4, 18, 0, 21, 4, 14, 0, 23, 0, 18, 9, 15, 0, 28, 0, 34, 9, 14, 0, 26, 4, 20, 8

Offset

10,1

Links

Table of n, a(n) for n=10..87.

Example

a(20) = 6 counts these triples: (2,1,12), (2,2,6), (2,3,4), (2,4,3), (2,6,2), (2,12,1).

Mathematica

t[n_, c_] := Module[{r}, r = Flatten[Table[If[n - 2 x^2 <= 0, {},
Map[({x, #, Quotient[n - 2 x^2, #]} &),
Select[Divisors[n - 2 x^2], Divisible[n - 2 x^2, #] &]]],
{x, 1, Floor[Sqrt[n - 1]]}], 1]; Select[r, Apply[c, #] &]];
c = (! CoprimeQ[#1, #2 #3] &);
Table[Length[t[n, c]], {n, 10, 130}]
(* Peter J. C. Moses, Mar 29 2026 *)

Crossrefs

Cf. A393710, A394038, A394788, A395634.

Sequence in context: A338569 A343757 A108760 * A343554 A137304 A234581

Adjacent sequences: A395632 A395633 A395634 * A395636 A395637 A395638

Keyword

nonn

Author

Clark Kimberling, May 05 2026

Status

approved