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

1,1

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.

Links

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

Formula

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

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

Example

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.

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<a then print(a+b+c); end if;
end do;
end do;

Prog

(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-b<a), listput(list, a+b+c)); ); ); Vec(list); }; \\ Michel Marcus, May 10 2021

Crossrefs

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

Cf. A020886, A020890.

Sequence in context: A063913 A119705 A343895 * A155560 A353198 A217734

Adjacent sequences: A343891 A343892 A343893 * A343895 A343896 A343897

Keyword

nonn

Author

Bernard Schott, May 07 2021

Status

approved