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A339858 Middle side of integer-sided primitive triangles whose sides a < b < c form a geometric progression. 4
6, 12, 20, 30, 35, 40, 42, 56, 63, 70, 77, 72, 88, 90, 99, 117, 126, 110, 130, 132, 143, 154, 165, 176, 187, 156, 204, 228, 182, 195, 208, 221, 234, 247, 260, 273, 210, 238, 266, 240, 255, 285, 330, 345, 272, 304, 336, 368, 400, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

This sequence is not increasing. For example, a(11) = 77 for triple (49, 77, 121) while a(12) = 72 for triple (64, 72, 81).

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).

LINKS

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

Project Euler, Problem 370: Geometric triangles.

FORMULA

a(n) = A339856 (n, 2).

EXAMPLE

a(1) = 6 only for the smallest such triangle (4, 6, 9) with 6^2 = 4*9 and a ratio q = 3/2.

a(2) = 12 only for the triangle (9, 12, 16) with 12^2 = 9*16 and a ratio q = 4/3.

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(b); 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(b, ", "); ); ); ); ); } \\ Michel Marcus, Dec 30 2020

CROSSREFS

Cf. A339856 (triples), A339857 (smallest side), this sequence (middle side), A339859 (largest side), A339860 (perimeter).

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

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

Cf. A002378 \ {0,2} (a subsequence).

Sequence in context: A079760 A109895 A083209 * A080714 A116368 A343065

Adjacent sequences:  A339855 A339856 A339857 * A339859 A339860 A339862

KEYWORD

nonn

AUTHOR

Bernard Schott, Dec 29 2020

STATUS

approved

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Last modified April 15 00:13 EDT 2021. Contains 342971 sequences. (Running on oeis4.)