A339857 Smallest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

4, 9, 16, 25, 25, 25, 36, 49, 49, 49, 49, 64, 64, 81, 81, 81, 81, 100, 100, 121, 121, 121, 121, 121, 121, 144, 144, 144, 169, 169, 169, 169, 169, 169, 169, 169, 196, 196, 196, 225, 225, 225, 225, 225, 256, 256, 256, 256, 256, 289, 289, 289, 289, 289, 289, 289, 289, 289, 289

Offset: 1

Author

Bernard Schott, Dec 25 2020

Keywords

• nonn

Comments

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

All the terms are the squares >= 4 in increasing order.

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

Examples

			a(1) = 4 for only the smallest such triangle (4, 6, 9).
a(4) = 25 for triple (25, 30, 36) with 25 * 36 = 30^2 and ratio q_1 = 6/5, hence for this triangle, C < Pi/2 because 1 < q_1 = 6/5 < sqrt(phi); also a(5) = 25 for the triple (25, 35, 49) with 25 * 49 = 35^2 and ratio q_2 = 7/5; then a(6) = 25 for the triple (25, 40, 64) with 25*64 = 40^2 and ratio q_3 = 8/5, hence, for these two last triangles, C > Pi/2 because sqrt(phi) < q_2 < q_3 < phi.
		

Links

• Project Euler, Problem 370: Geometric Triangles.

Crossrefs

Cf. A339856 (triples), this sequence (smallest side), A339858 (middle side), A339859 (largest side), A339860 (perimeter).

Cf. A336751 (similar for sides in arithmetic progression).

Cf. A335894 (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(a); 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(a, ", ")); ); ); ); } \\ Michel Marcus, Dec 26 2020

Formula

a(n) = A339856(n, 1).