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A007432 Moebius transform applied thrice to natural numbers.
(Formerly M0031)
2

%I M0031

%S 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 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 -8,16,4,0,-26,56,0,58,-28,12,8,20,0,64,14

%N Moebius transform applied thrice to natural numbers.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007432/b007432.txt">Table of n, a(n) for n=1..1000</a>

%H 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 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 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Multiplicative with a(p^e) = sum (-1)^k (3 choose k)p^(e-k)[e>=k], k=0..3

%F Dirichlet g.f.: zeta(x-1)/zeta^3(x)

%F 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} A007427. - _Enrique Pérez Herrero_, Jan 19 2013

%F Sum_{k=1..n} a(k) ~ 108 * n^2 / Pi^6. - _Vaclav Kotesovec_, Nov 04 2018

%t 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 Cf. A007431.

%K sign,easy,nice,mult

%O 1,5

%A _N. J. A. Sloane_.

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)