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

1,1

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.

Links

Table of n, a(n) for n=1..59.

Project Euler, Problem 370: Geometric Triangles.

Formula

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

Example

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.

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;

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

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).

Sequence in context: A259602 A017668 A225004 * A074373 A067115 A061077

Adjacent sequences: A339854 A339855 A339856 * A339858 A339859 A339860

Keyword

nonn

Author

Bernard Schott, Dec 25 2020

Status

approved