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

%I A123323 #37 Nov 23 2024 14:26:35
%S A123323 1,1,3,4,8,7,15,14,21,20,35,26,48,39,52,52,80,57,99,76,102,95,143,100,
%T A123323 160,132,171,150,224,148,255,200,250,224,300,222,360,279,348,296,440,
%U A123323 294,483,370,444,407,575,392,609,460,592,516,728,495,740,588,738,644
%N A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.
%C A123323 Number of triples a,b,c with a <= b <= c < a+b, gcd(a,b,c) = 1 and c = n.
%C A123323 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
%H A123323 Alois P. Heinz, <a href="/A123323/b123323.txt">Table of n, a(n) for n = 1..10000</a>
%H A123323 Stackexchange, <a href="http://math.stackexchange.com/questions/689611/number-of-triples-for-which-gcda-b-c-1-and-c-n">Number of triples for which gcd(a,b,c)=1 and c=n</a>, Feb 25 2014
%F A123323 Moebius transform of b(n) = floor((n+1)^2/4).
%F A123323 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)).
%p A123323 with(numtheory):
%p A123323 a:= n-> add(mobius(n/d)*floor((d+1)^2/4), d=divisors(n)):
%p A123323 seq(a(n), n=1..60);  # _Alois P. Heinz_, Oct 23 2013
%t A123323 a[n_] := DivisorSum[n, Floor[(#+1)^2/4]*MoebiusMu[n/#]&]; Array[a, 60] (* _Jean-François Alcover_, Dec 07 2015 *)
%o A123323 (PARI) A123323(n)=sumdiv(n,d,floor((d+1)^2/4)*moebius(n/d))
%Y A123323 Cf. A002620, A051493, A054875, A070110, A123324, A123325, A239246.
%K A123323 easy,look,nonn
%O A123323 1,3
%A A123323 _Franklin T. Adams-Watters_, Sep 25 2006