login
A123323
Number of integer-sided triangles with maximum side n, with sides relatively prime.
11
1, 1, 3, 4, 8, 7, 15, 14, 21, 20, 35, 26, 48, 39, 52, 52, 80, 57, 99, 76, 102, 95, 143, 100, 160, 132, 171, 150, 224, 148, 255, 200, 250, 224, 300, 222, 360, 279, 348, 296, 440, 294, 483, 370, 444, 407, 575, 392, 609, 460, 592, 516, 728, 495, 740, 588, 738, 644
OFFSET
1,3
COMMENTS
Number of triples a,b,c with a <= b <= c < a+b, gcd(a,b,c) = 1 and c = n.
Dropping the requirement for side lengths to be relatively prime this sequence becomes A002620 (with a different offset). See the Sep 2006 comment in A002620. - Peter Munn, Jul 26 2017
FORMULA
Moebius transform of b(n) = floor((n+1)^2/4).
G.f.: (G(x)+x-x^2)/2, where G(x) = Sum_{k >= 1} mobius(k)*x^k*(1+2*x^k-x^(2*k))/(1-x^k)^2/(1-x^(2*k)).
MAPLE
with(numtheory):
a:= n-> add(mobius(n/d)*floor((d+1)^2/4), d=divisors(n)):
seq(a(n), n=1..60); # Alois P. Heinz, Oct 23 2013
MATHEMATICA
a[n_] := DivisorSum[n, Floor[(#+1)^2/4]*MoebiusMu[n/#]&]; Array[a, 60] (* Jean-François Alcover, Dec 07 2015 *)
PROG
(PARI) A123323(n)=sumdiv(n, d, floor((d+1)^2/4)*moebius(n/d)).
KEYWORD
easy,look,nonn
AUTHOR
STATUS
approved