A343894 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c. The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide, in increasing order of side b.

13, 37, 47, 71, 73, 107, 121, 143, 183, 177, 181, 191, 241, 239, 249, 253, 291, 299, 347, 337, 359, 409, 421, 429, 431, 433, 491, 517, 503, 529, 563, 537, 541, 579, 587, 649, 659, 661, 671, 753, 743, 769, 759, 781, 831, 767, 789, 793, 897, 851, 923, 863, 913, 947, 1033, 933

Offset: 1

Author

Bernard Schott, May 07 2021

Keywords

• nonn

Comments

The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.

The sequence is not monotonic: a(9) = 183 > a(10) = 177.

All terms are odd.

For the corresponding primitive triples and miscellaneous properties and references, see A343891.

Examples

			a(3) = 15 + 12 + 20 = 47, because the third triple is (15, 12, 20) with relations 2/15 = 1/12 + 1/20 and 20-15 < 12 < 20+15.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343895.

Cf. A020886, A020890.

Programs

Maple

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b
				

PARI

lista(nn) = {my(list = List()); for (a=4, nn, for (b = floor(a/2)+1, a-1, my(c = a*b/(2*b-a)); if ((denominator(c) == 1) && (gcd([a, b, c]) == 1) && (c-bMichel Marcus, May 10 2021

Formula

a(n) = A343891(n, 1) + A343891(n, 2) + A343891(n, 3).

a(n) = A020883(n) + A343892(n) + A343893(n).