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

Author

Bernard Schott, Jan 08 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

• Project Euler, Problem 370: Geometric triangles.

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.

Programs

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;

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

Formula

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