|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|