A394785 - OEIS

%I #10 Apr 28 2026 17:26:31

%S 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,

%T 15,10,16,15,18,21,12,20,23,12,18,23,19,14,20,20,19,24,12,18,28,16,24,

%U 29,23,20,28,22,28,27,20,24,37,14,20,36,28,20,30,19,30

%N 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.

%H Robert Israel, <a href="/A394785/b394785.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = Sum_{1 <= x <= floor(sqrt(n/2))} A000005(n-x^2). - _Robert Israel_, Apr 23 2026

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

%p f:= proc(n) local x;

%p       add(NumberTheory:-tau(n-x^2),x=1..floor(sqrt(n/2)))

%p end proc:

%p map(f, [$0..100]); # _Robert Israel_, Apr 23 2026

%t t[n_, c_] := Module[{r}, r = Flatten[Table[If[n - x^2 <= 0, {},

%t        Map[({x, #, Quotient[n - x^2, #]} &),

%t        Select[Divisors[n - x^2], Divisible[n - x^2, #] &]]], {x, 1,

%t        Floor[Sqrt[n - 1]]}], 1]; Select[r, Apply[c, #] &]];

%t c = ((#1)^2 <= #2*#3 &);

%t Join[{0}, Table[Length[t[n, c]], {n, 1, 130}]]

%t (* _Peter J. C. Moses_, Mar 29 2026 *)

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

%K nonn

%O 0,4

%A _Clark Kimberling_, Apr 16 2026