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A007438 Moebius transform of triangular numbers.

%I A007438 M1339 #48 Jul 08 2025 17:04:20
%S A007438 1,2,5,7,14,13,27,26,39,38,65,50,90,75,100,100,152,111,189,148,198,
%T A007438 185,275,196,310,258,333,294,434,292,495,392,490,440,588,438,702,549,
%U A007438 684,584,860,582,945,730,876,803,1127,776,1197,910,1168,1020,1430
%N A007438 Moebius transform of triangular numbers.
%C A007438 a(n)=|{(x,y):1<=x<=y<=n, gcd(x,y,n)=1}|. E.g. a(4)=7 because of the pairs (1,1), (1,2), (1,3), (1,4), (2,3), (3,3), (3,4). - _Steve Butler_, Apr 18 2006
%C A007438 Partial sums of a(n) give A015631(n). - _Steve Butler_, Apr 18 2006
%C A007438 Equals row sums of triangle A159905. - _Gary W. Adamson_, Apr 25 2009; corrected by _Mats Granvik_, Apr 24 2010
%D A007438 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007438 Alois P. Heinz, <a href="/A007438/b007438.txt">Table of n, a(n) for n = 1..10000</a>
%H A007438 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A007438 a(n) = (A007434(n)+A000010(n))/2, half the sum of the Mobius transforms of n^2 and n. Dirichlet g.f. (zeta(s-2)+zeta(s-1))/(2*zeta(s)). - _R. J. Mathar_, Feb 09 2011
%F A007438 G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/(1 - x)^3. - _Ilya Gutkovskiy_, Apr 25 2017
%p A007438 with(numtheory):
%p A007438 a:= proc(n) option remember;
%p A007438        add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
%p A007438     end:
%p A007438 seq(a(n), n=1..60);  # _Alois P. Heinz_, Feb 09 2011
%t A007438 a[n_] := Sum[MoebiusMu[n/d]*d*(d+1)/2, {d, Divisors[n]}]; Array[a, 60] (* _Jean-François Alcover_, Apr 17 2014 *)
%o A007438 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*(d+1)/2); \\ _Michel Marcus_, Nov 05 2018
%Y A007438 Cf. A000217.
%Y A007438 Cf. A159905. - _Gary W. Adamson_, Apr 25 2009
%K A007438 nonn
%O A007438 1,2
%A A007438 _N. J. A. Sloane_