

A339858


Middle side of integersided 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
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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



