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

19, 37, 61, 91, 109, 129, 127, 169, 193, 219, 247, 217, 273, 271, 301, 367, 403, 331, 399, 397, 433, 471, 511, 553, 597, 469, 637, 733, 547, 589, 633, 679, 727, 777, 829, 883, 631, 723, 823, 721, 769, 871, 1039, 1099, 817, 921, 1033, 1153, 1281, 919, 973, 1029, 1087

Offset

1,1

Comments

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order.

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).

For the corresponding primitive triples and miscellaneous properties, see A339859.

Links

Table of n, a(n) for n=1..53.

Project Euler, Problem 370: Geometric triangles.

Formula

a(n) = A339856(n, 1) + A339856(n, 2) + A339856(n, 3).

Example

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.
a(2) = 37 = 9+12+16 for the triple (9, 12, 16) with 9 * 16 = 12^2 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(a+b+c); 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(a+b+c, ", ")); ); ); ); } \\ Michel Marcus, Jan 08 2021

Crossrefs

Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), A339859 (largest side), this sequence (perimeter).

Cf. A336754 (similar for sides in arithmetic progression).

Cf. A335897 (similar for angles in arithmetic progression).

Subsequence of A034017.

Sequence in context: A175546 A162471 A124268 * A259648 A154944 A039321

Adjacent sequences: A339857 A339858 A339859 * A339861 A339862 A339863

Keyword

nonn

Author

Bernard Schott, Jan 08 2021

Status

approved