A339858 Middle side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

6, 12, 20, 30, 35, 40, 42, 56, 63, 70, 77, 72, 88, 90, 99, 117, 126, 110, 130, 132, 143, 154, 165, 176, 187, 156, 204, 228, 182, 195, 208, 221, 234, 247, 260, 273, 210, 238, 266, 240, 255, 285, 330, 345, 272, 304, 336, 368, 400, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459

Offset: 1

Author

Bernard Schott, Dec 29 2020

Keywords

• nonn

Comments

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

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

This sequence is not increasing. For example, a(11) = 77 for triple (49, 77, 121) while a(12) = 72 for triple (64, 72, 81).

Oblong numbers k*(k+1) >= 6 form a subsequence (A002378) and belong to triples of the form (k^2, k*(k+1), (k+1)^2).

Examples

			a(1) = 6 only for the smallest such triangle (4, 6, 9) with 6^2 = 4*9 and a ratio q = 3/2.
a(2) = 12 only for the triangle (9, 12, 16) with 12^2 = 9*16 and a ratio q = 4/3.
		

Links

• Project Euler, Problem 370: Geometric triangles.

Crossrefs

Cf. A339856 (triples), A339857 (smallest side), this sequence (middle side), A339859 (largest side), A339860 (perimeter).

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

Cf. A335894 (similar for angles in arithmetic progression).

Cf. A002378 \ {0,2} (a subsequence).

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(b); 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(b, ", "); ); ); ); );} \\ Michel Marcus, Dec 30 2020

Formula

a(n) = A339856 (n, 2).