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

1,30

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

Links

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

Example

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.

Mathematica

a[n_] := Count[{x, y, z} /. {ToRules[Reduce[x>0 && y>0 && z>0 && z<y && x*y*(y+z) == n, {x, y, z}, Integers]]}, {x_, y_, z_} /; CoprimeQ[y, z] && Not[OddQ[y] && OddQ[z] || EvenQ[y] && EvenQ[z]]]; Array[a, 117] (* Jean-François Alcover, Jun 03 2017 *)

Prog

(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

Crossrefs

Cf. A010814. Nonzero for terms in A005279.

Sequence in context: A128616 A331902 A333817 * A353333 A353303 A307666

Adjacent sequences: A270414 A270415 A270416 * A270418 A270419 A270420

Keyword

nonn

Author

Henry Bottomley, Mar 16 2016

Status

approved