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A007432 Moebius transform applied thrice to natural numbers.

%I A007432 M0031 #36 Dec 17 2021 11:28:00
%S A007432 1,-1,0,1,2,0,4,1,3,-2,8,0,10,-4,0,2,14,-3,16,2,0,-8,20,0,13,-10,8,4,
%T A007432 26,0,28,4,0,-14,8,3,34,-16,0,2,38,0,40,8,6,-20,44,0,31,-13,0,10,50,
%U A007432 -8,16,4,0,-26,56,0,58,-28,12,8,20,0,64,14
%N A007432 Moebius transform applied thrice to natural numbers.
%D A007432 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007432 T. D. Noe, <a href="/A007432/b007432.txt">Table of n, a(n) for n=1..1000</a>
%H A007432 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A007432 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A007432 Vaclav Kotesovec, <a href="/A007432/a007432.jpg">Plot of Sum_{k=1..n} a(k) / (108*n^2/Pi^6) for n = 1..1000000</a>
%H A007432 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A007432 Multiplicative with a(p^e) = Sum_{k=0..3} (-1)^k C(3,k)*p^(e-k)[e>=k];
%F A007432 Dirichlet g.f.: zeta(s-1)/zeta^3(s).
%F A007432 a(n) = Sum{d|n} tau_{-3}(d)*n/d = Sum{d|n} tau_{-2}(d)*phi(n/d), where tau_{-3} is A007428 and tau_{-2} is A007427. - _Enrique Pérez Herrero_, Jan 19 2013
%F A007432 Sum_{k=1..n} a(k) ~ 108 * n^2 / Pi^6. - _Vaclav Kotesovec_, Nov 04 2018
%t A007432 a[p_, e_] := Sum[ (-1)^k*Binomial[3, k]*p^(e - k), {k, 0, Min[e, 3]}]; a[n_] := Times @@ Apply[a, FactorInteger[n], {1}]; a[1] = 1; Table[ a[n], {n, 1, 68}] (* _Jean-François Alcover_, Dec 28 2011, after formula *)
%Y A007432 Cf. A007431.
%K A007432 sign,easy,nice,mult
%O A007432 1,5
%A A007432 _N. J. A. Sloane_