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A339860 Perimeter of primitive integer-sided triangles whose sides a < b < c form a geometric progression.

%I A339860 #17 Jan 11 2021 14:41:23
%S A339860 19,37,61,91,109,129,127,169,193,219,247,217,273,271,301,367,403,331,
%T A339860 399,397,433,471,511,553,597,469,637,733,547,589,633,679,727,777,829,
%U A339860 883,631,723,823,721,769,871,1039,1099,817,921,1033,1153,1281,919,973,1029,1087
%N A339860 Perimeter of primitive integer-sided triangles whose sides a < b < c form a geometric progression.
%C A339860 The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order.
%C A339860 These perimeters are of the form r^2 + r*s + s^2, r < s, gcd(r, s) = 1 and q = r/s (A034017), so they are all odd but not in increasing order. For example, a(6) = 129 for triple (25, 40, 64) while a(7) = 127 for triple (36, 42, 49).
%C A339860 For the corresponding primitive triples and miscellaneous properties, see A339859.
%H A339860 Project Euler, <a href="https://projecteuler.net/problem=370">Problem 370: Geometric triangles</a>.
%F A339860 a(n) = A339856(n, 1) + A339856(n, 2) + A339856(n, 3).
%e A339860 a(1) = 19 = 4+6+9 for the smallest such triangle (4, 6, 9) with 4 * 9 = 6^2 and a ratio q = 3/2.
%e A339860 a(2) = 37 = 9+12+16 for the triple (9, 12, 16) with 9 * 16 = 12^2 and a ratio q = 4/3.
%p A339860 for a from 1 to 300 do
%p A339860 for b from a+1 to floor((1+sqrt(5))/2 *a) do
%p A339860 for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
%p A339860 if k=b^2 and igcd(a, b, c)=1 then print(a+b+c); end if;
%p A339860 end do;
%p A339860 end do;
%p A339860 end do;
%o A339860 (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+b+c, ", ")); ); ); ); } \\ _Michel Marcus_, Jan 08 2021
%Y A339860 Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), A339859 (largest side), this sequence (perimeter).
%Y A339860 Cf. A336754 (similar for sides in arithmetic progression).
%Y A339860 Cf. A335897 (similar for angles in arithmetic progression).
%Y A339860 Subsequence of A034017.
%K A339860 nonn
%O A339860 1,1
%A A339860 _Bernard Schott_, Jan 08 2021