A343894 - OEIS

%I #25 May 19 2021 16:06:46

%S 13,37,47,71,73,107,121,143,183,177,181,191,241,239,249,253,291,299,

%T 347,337,359,409,421,429,431,433,491,517,503,529,563,537,541,579,587,

%U 649,659,661,671,753,743,769,759,781,831,767,789,793,897,851,923,863,913,947,1033,933

%N 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.

%C 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.

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

%C All terms are odd.

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

%H Michel Marcus, <a href="/A343894/b343894.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%e 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.

%p for a from 4 to 200 do

%p for b from floor(a/2)+1 to a-1 do

%p c := a*b/(2*b-a);

%p if c=floor(c) and igcd(a,b,c)=1 and c-b<a then print(a+b+c); end if;

%p end do;

%p end do;

%o (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

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

%Y Cf. A020886, A020890.

%K nonn

%O 1,1

%A _Bernard Schott_, May 07 2021