A270417 Number of integer-sided right triangles with semiperimeter n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1

Offset: 1

Author

Henry Bottomley, Mar 16 2016

Keywords

• nonn

Comments

Number of positive integer solutions to x*y*(y+z) = n with y and z coprime and of opposite parity and z < y.

Records occur at 1, 6, 30, 60, 120, 210, 360, 420, 840, 1260, 2310, 2520, 4620, 9240, 13860, 27720, 55440, 60060, ... - Antti Karttunen, Sep 25 2018

Examples

			a(25)=0 since 2*25 = 50 is not the perimeter of a suitable triangle;
a(30)=2 since 2*30 = 60 = 15+20+25 = 10+24+26;
a(35)=1 since 2*35 = 70 = 20+21+29.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A010814. Nonzero for terms in A005279.

Programs

Mathematica

a[n_] := Count[{x, y, z} /. {ToRules[Reduce[x>0 && y>0 && z>0 && zJean-François Alcover, Jun 03 2017 *)

PARI

A270417(n) = { my(s=0); fordiv(n,x,fordiv(n/x,y,my(w=n/(x*y)); if((w < 2*y)&&(w>y)&&(w%2)&&(1==gcd(w,y)),s++))); (s); }; \\ (Here z = w-y) - Antti Karttunen, Sep 25 2018