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

1,3

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

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

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)).

Crossrefs

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

Sequence in context: A049826 A310014 A310015 * A034772 A197138 A065309

Adjacent sequences: A123320 A123321 A123322 * A123324 A123325 A123326

Keyword

easy,look,nonn

Author

Franklin T. Adams-Watters, Sep 25 2006

Status

approved