%I #24 Feb 21 2021 10:45:45

%S 4,9,16,25,25,25,36,49,49,49,49,64,64,81,81,81,81,100,100,121,121,121,

%T 121,121,121,144,144,144,169,169,169,169,169,169,169,169,196,196,196,

%U 225,225,225,225,225,256,256,256,256,256,289,289,289,289,289,289,289,289,289,289

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

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

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

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

%H Project Euler, <a href="https://projecteuler.net/problem=370">Problem 370: Geometric Triangles</a>.

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

%e a(1) = 4 for only the smallest such triangle (4, 6, 9).

%e 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.

%p for a from 1 to 300 do

%p for b from a+1 to floor((1+sqrt(5))/2 *a) do

%p for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;

%p if k=b^2 and igcd(a,b,c)=1 then print(a); end if;

%p end do;

%p end do;

%p end do;

%o (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

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

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

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

%K nonn

%O 1,1

%A _Bernard Schott_, Dec 25 2020