A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

9, 16, 25, 36, 49, 64, 49, 64, 81, 100, 121, 81, 121, 100, 121, 169, 196, 121, 169, 144, 169, 196, 225, 256, 289, 169, 289, 361, 196, 225, 256, 289, 324, 361, 400, 441, 225, 289, 361, 256, 289, 361, 484, 529, 289, 361, 441, 529, 625, 324, 361, 400, 441, 484, 529, 576

Offset

1,1

Comments

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order. This sequence lists the c's.

For the corresponding primitive triples and miscellaneous properties and references, see A339856.

The terms are all squares >= 9 but they are not in increasing order. For example, a(6) = 64 for triple (25, 40, 64) while a(7) = 49 for triple (36, 42, 49).

Links

Table of n, a(n) for n=1..56.

Project Euler, Problem 370: Geometric triangles.

Formula

a(n) = A339856(n, 3).

Example

a(1) = 9 for only the smallest such triangle (4, 6, 9) with 6^2 = 4*9, this one corresponds to an obtuse triangle because sqrt(phi) < q = 3/2 < phi, hence C > Pi/2.
a(3) = 25 for only the triple (16, 20, 25) with 16 * 25 = 20^2, this one corresponds to an acute triangle because 1 < q = 5/4 < sqrt(phi), hence C < Pi/2.

Maple

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 *a) do
for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(c); end if;
end do;
end do;
end do;

Prog

(PARI) lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print1(c, ", "); ); ); ); ); } \\ Michel Marcus, Jan 07 2021

Crossrefs

Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), this sequence (largest side), A339860 (perimeter).

Cf. A336753 (similar for sides in arithmetic progression).

Cf. A335896 (similar for angles in arithmetic progression).

Sequence in context: A151973 A102219 A227650 * A076093 A085585 A026062

Adjacent sequences: A339856 A339857 A339858 * A339860 A339861 A339862

Keyword

nonn

Author

Bernard Schott, Jan 05 2021

Status

approved