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Number of integer-sided right triangles with semiperimeter n.
1

%I #18 Dec 10 2019 23:16:15

%S 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,

%T 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,

%U 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

%N Number of integer-sided right triangles with semiperimeter n.

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

%C 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

%H Antti Karttunen, <a href="/A270417/b270417.txt">Table of n, a(n) for n = 1..65537</a>

%e a(25)=0 since 2*25 = 50 is not the perimeter of a suitable triangle;

%e a(30)=2 since 2*30 = 60 = 15+20+25 = 10+24+26;

%e a(35)=1 since 2*35 = 70 = 20+21+29.

%t 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 *)

%o (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

%Y Cf. A010814. Nonzero for terms in A005279.

%K nonn

%O 1,30

%A _Henry Bottomley_, Mar 16 2016