A394785 a(n) = number of triples (x, y, z) such that x^2 + y*z = n, where x,y,z are positive integers satisfying x^2 <= y*z.

0, 0, 1, 2, 2, 3, 2, 4, 5, 6, 7, 6, 6, 9, 6, 6, 10, 7, 9, 14, 9, 14, 12, 10, 12, 17, 9, 12, 14, 15, 10, 16, 15, 18, 21, 12, 20, 23, 12, 18, 23, 19, 14, 20, 20, 19, 24, 12, 18, 28, 16, 24, 29, 23, 20, 28, 22, 28, 27, 20, 24, 37, 14, 20, 36, 28, 20, 30, 19, 30

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

a(n) = Sum_{1 <= x <= floor(sqrt(n/2))} A000005(n-x^2). - Robert Israel, Apr 23 2026

Example

a(7) = 4 counts these triples: (1, 1, 6), (1, 2, 3), (1, 3, 2), (1, 6, 1).

Maple

f:= proc(n) local x;
      add(NumberTheory:-tau(n-x^2), x=1..floor(sqrt(n/2)))
end proc:
map(f, [$0..100]); # Robert Israel, Apr 23 2026

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 &);
Join[{0}, Table[Length[t[n, c]], {n, 1, 130}]]
(* Peter J. C. Moses, Mar 29 2026 *)

Crossrefs

Cf. A000005, A393710, A394740, A394743, A394786, A394787.

Sequence in context: A318839 A347664 A304105 * A305978 A305985 A319706

Adjacent sequences: A394782 A394783 A394784 * A394786 A394787 A394788

Keyword

nonn

Author

Clark Kimberling, Apr 16 2026

Status

approved