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

Author

Bernard Schott, Jan 05 2021

Keywords

• nonn

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).

Examples

			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.
		

Links

• Project Euler, Problem 370: Geometric triangles.

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).

Programs

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;

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

Formula

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