A393712 a(n) = number of triples (x, y, z) such that x^2 + y*z = n, where x, y, z are positive integers satisfying x < y <= z.

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 2, 2, 0, 3, 1, 2, 2, 1, 1, 5, 1, 1, 3, 4, 0, 4, 1, 3, 4, 1, 2, 6, 0, 3, 4, 5, 0, 4, 3, 4, 5, 1, 1, 8, 1, 4, 5, 4, 2, 4, 3, 5, 5, 3, 2, 9, 0, 2, 8, 6, 2, 6, 2, 6, 4, 4, 3, 9, 4, 2, 8, 3, 1, 9, 2, 10, 5, 1, 4, 10

Offset

0,14

Links

Table of n, a(n) for n=0..85.

Example

a(19) = 3 counts these triples: (1, 2, 9), (1, 3, 6), (2, 3, 5).

Mathematica

t[n_, c_] := Module[{r}, r = Flatten[Table[If[n - x^2 <= 0, {},
       Map[({x, #, Quotient[n - x^2, #]} &),
        Select[Divisors[n - x^2], Divisible[n - x^2, #] &]]], {x, 1,
       Floor[Sqrt[n - 1]]}], 1]; Select[r, Apply[c, #] &]];
c = (#1 < #2 && #2 <= #3 &); Table[{n, t[n, c]}, {n, 1, 30}]
Join[{0}, Table[Length[t[n, c]], {n, 1, 130}]]
(* Peter J. C. Moses, Mar 31 2026 *)

Crossrefs

Cf. A393710, A393711, A394746.

Sequence in context: A147755 A307195 A231188 * A249695 A136748 A235919

Adjacent sequences: A393709 A393710 A393711 * A393713 A393714 A393715

Keyword

nonn

Author

Clark Kimberling, Mar 31 2026

Status

approved