A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.

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

Author

Franklin T. Adams-Watters, Sep 25 2006

Keywords

• easy
• look
• nonn

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Stackexchange, Number of triples for which gcd(a,b,c)=1 and c=n, Feb 25 2014

Crossrefs

Cf. A002620, A051493, A054875, A070110, A123324, A123325, A239246.

Programs

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 *)

PARI

A123323(n)=sumdiv(n,d,floor((d+1)^2/4)*moebius(n/d))

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)).