A077177 - OEIS

%I #20 Apr 24 2022 17:19:54

%S 0,0,1,0,1,2,3,5,8,17,34,59,111,213,396,746,1413,2690,5147,9826,18885,

%T 36269,69952,134949,260743,504636,978311,1899832,3692980,7190329,

%U 13994206,27279898,53195986

%N Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.

%C A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

%C Equivalently, number of divisors of s=A070826(n) in the range (sqrt(s), sqrt(2s)). More generally, for any positive integer s, the number of primitive Pythagorean triangles with perimeter 2's equals the number of odd unitary divisors of s in the range (sqrt(s), sqrt(2s)). (A divisor d of n is 'unitary' if gcd(d, n/d) = 1.)

%D A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.

%D Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.

%D F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), p. 52.

%H Randall L. Rathbun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;90df5cf0.0206">Equal Perimeter primitive right triangles</a>

%F a(n) = A070109(A002110(n)) = A078926(A070826(n)).

%e a(5) = 1 since there is exactly one primitive Pythagorean triangle with perimeter 2*3*5*7*11; its edge lengths are (132, 1085, 1093). a(7) = 3; the 3 triangles have edge lengths (70941, 214060, 225509), (96460, 195789, 218261) and (142428, 156485, 211597).

%t a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]

%o (PARI) semi_peri(p)= {local(q,r,ct,tot); ct=0; tot=0; pt=0; fordiv(p,q,r=p/q-q; if(r<=q&&r>0,print(q,",",r," [",gcd(q,r),"] "); if(gcd(q,r)==1,ct=ct+1; if(q*r%2==0,pt=pt+1; ); ); tot=tot+1); ); print("semiperimeter:"p," Total sets:",tot," Coprime:",ct," Primitive:",pt); } /* Lists all pairs q,r such that the triangle with edge lengths (q^2-r^2, 2qr, q^2+r^2) has semiperimeter p. */

%Y Cf. A002110, A070109, A070826, A078926.

%K more,nonn

%O 1,6

%A Kermit Rose and _Randall L Rathbun_, Nov 29 2002

%E Edited by _Dean Hickerson_, Dec 18 2002