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A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

%I A339859 #22 Feb 21 2021 13:53:16
%S A339859 9,16,25,36,49,64,49,64,81,100,121,81,121,100,121,169,196,121,169,144,
%T A339859 169,196,225,256,289,169,289,361,196,225,256,289,324,361,400,441,225,
%U A339859 289,361,256,289,361,484,529,289,361,441,529,625,324,361,400,441,484,529,576
%N A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.
%C A339859 The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order. This sequence lists the c's.
%C A339859 For the corresponding primitive triples and miscellaneous properties and references, see A339856.
%C A339859 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).
%H A339859 Project Euler, <a href="https://projecteuler.net/problem=370">Problem 370: Geometric triangles</a>.
%F A339859 a(n) = A339856(n, 3).
%e A339859 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.
%e A339859 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.
%p A339859 for a from 1 to 300 do
%p A339859 for b from a+1 to floor((1+sqrt(5))/2 *a) do
%p A339859 for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
%p A339859 if k=b^2 and igcd(a, b, c)=1 then print(c); end if;
%p A339859 end do;
%p A339859 end do;
%p A339859 end do;
%o A339859 (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
%Y A339859 Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), this sequence (largest side), A339860 (perimeter).
%Y A339859 Cf. A336753 (similar for sides in arithmetic progression).
%Y A339859 Cf. A335896 (similar for angles in arithmetic progression).
%K A339859 nonn
%O A339859 1,1
%A A339859 _Bernard Schott_, Jan 05 2021