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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. 8
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 (list; graph; refs; listen; history; text; internal format)
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 A217734 A319969

Adjacent sequences:  A343891 A343892 A343893 * A343895 A343896 A343897

KEYWORD

nonn

AUTHOR

Bernard Schott, May 07 2021

STATUS

approved

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Last modified June 23 23:28 EDT 2021. Contains 345403 sequences. (Running on oeis4.)