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%I #22 Feb 21 2021 10:42:03
%S 6,12,20,30,35,40,42,56,63,70,77,72,88,90,99,117,126,110,130,132,143,
%T 154,165,176,187,156,204,228,182,195,208,221,234,247,260,273,210,238,
%U 266,240,255,285,330,345,272,304,336,368,400,306,323,340,357,374,391,408,425,442,459
%N Middle 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 b's.
%C For the corresponding primitive triples and miscellaneous properties and references, see A339856.
%C This sequence is not increasing. For example, a(11) = 77 for triple (49, 77, 121) while a(12) = 72 for triple (64, 72, 81).
%C 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).
%H Project Euler, <a href="https://projecteuler.net/problem=370">Problem 370: Geometric triangles</a>.
%F a(n) = A339856 (n, 2).
%e a(1) = 6 only for the smallest such triangle (4, 6, 9) with 6^2 = 4*9 and a ratio q = 3/2.
%e a(2) = 12 only for the triangle (9, 12, 16) with 12^2 = 9*16 and a ratio q = 4/3.
%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(b); 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(b, ", "); ); ); ); );} \\ _Michel Marcus_, Dec 30 2020
%Y Cf. A339856 (triples), A339857 (smallest side), this sequence (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).
%Y Cf. A002378 \ {0,2} (a subsequence).
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 29 2020
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