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A077177 Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes. 0
0, 0, 1, 0, 1, 2, 3, 5, 8, 17, 34, 59, 111, 213, 396, 746, 1413, 2690, 5147, 9826, 18885, 36269, 69952, 134949, 260743, 504636, 978311, 1899832, 3692980, 7190329, 13994206, 27279898, 53195986 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.
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.)
REFERENCES
A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.
Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.
F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), p. 52.
LINKS
FORMULA
a(n) = A070109(A002110(n)) = A078926(A070826(n)).
EXAMPLE
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).
MATHEMATICA
a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]
PROG
(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. */
CROSSREFS
Sequence in context: A108054 A342690 A123612 * A303874 A145793 A113879
KEYWORD
more,nonn
AUTHOR
Kermit Rose and Randall L Rathbun, Nov 29 2002
EXTENSIONS
Edited by Dean Hickerson, Dec 18 2002
STATUS
approved

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Last modified June 22 21:37 EDT 2024. Contains 373610 sequences. (Running on oeis4.)