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A007428 Moebius transform applied thrice to sequence 1,0,0,0,....

%I A007428 M2271 #52 Mar 18 2022 00:30:08
%S A007428 1,-3,-3,3,-3,9,-3,-1,3,9,-3,-9,-3,9,9,0,-3,-9,-3,-9,9,9,-3,3,3,9,-1,
%T A007428 -9,-3,-27,-3,0,9,9,9,9,-3,9,9,3,-3,-27,-3,-9,-9,9,-3,0,3,-9,9,-9,-3,
%U A007428 3,9,3,9,9,-3,27,-3,9,-9,0,9,-27,-3,-9,9,-27,-3,-3,-3,9,-9,-9,9,-27
%N A007428 Moebius transform applied thrice to sequence 1,0,0,0,....
%C A007428 Dirichlet inverse of A007425. - _R. J. Mathar_, Jul 15 2010
%C A007428 abs(a(n)) is the number of ways to write n=xyz where x,y,z are squarefree numbers. - _Benoit Cloitre_, Jan 02 2018
%D A007428 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007428 Enrique Pérez Herrero, <a href="/A007428/b007428.txt">Table of n, a(n) for n = 1..10000</a>
%H A007428 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A007428 Multiplicative with a(p^e) = (3 choose e) (-1)^e.
%F A007428 Dirichlet g.f.: 1/zeta(s)^3.
%F A007428 From _Enrique Pérez Herrero_, Jul 12 2010: (Start)
%F A007428 a(n^3) = A008683(n).
%F A007428 a(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End)
%F A007428 a(A046101(n)) = 0. - _Enrique Pérez Herrero_, Sep 07 2017
%F A007428 a(n) = Sum_{a*b*c=n} mu(a)*mu(b)*mu(c). - _Benedict W. J. Irwin_, Mar 02 2022
%p A007428 möbius := proc(a)  local b, i, mo: b := NULL:
%p A007428 mo := (m,n) -> `if`(irem(m,n) = 0, numtheory:-mobius(m/n), 0);
%p A007428 for i to nops(a) do b := b, add(mo(i,j)*a[j], j=1..i) od: [b] end:
%p A007428 (möbius@@3)([1, seq(0, i=1..77)]); # _Peter Luschny_, Sep 08 2017
%t A007428 tau[1,n_Integer]:=1; SetAttributes[tau, Listable];
%t A007428 tau[k_Integer,n_Integer]:=Plus@@(tau[k-1,Divisors[n]])/; k > 1;
%t A007428 tau[k_Integer,n_Integer]:=Plus@@(tau[k+1,Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1;
%t A007428 A007428[n_]:=tau[ -3,n]; (* _Enrique Pérez Herrero_, Jul 12 2010 *)
%t A007428 a[n_] := Which[n==1, 1, PrimeQ[n], -3, True, Times @@ Map[Function[e, Binomial[3, e] (-1)^e], FactorInteger[n][[All, 2]]]];
%t A007428 Array[a, 100] (* _Jean-François Alcover_, Jun 20 2018 *)
%o A007428 (Haskell)
%o A007428 a007428 n = product
%o A007428    [a007318' 3 e * cycle [1,-1] !! fromIntegral e | e <- a124010_row n]
%o A007428 -- _Reinhard Zumkeller_, Oct 09 2013
%o A007428 (PARI) a(n) = {my(f=factor(n)); for (k=1, #f~, e = f[k,2]; f[k,1] = binomial(3, e)*(-1)^e; f[k,2] = 1); factorback(f);} \\ _Michel Marcus_, Jan 03 2018
%o A007428 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)^3)[n], ", ")) \\ _Vaclav Kotesovec_, Feb 22 2021
%Y A007428 Consecutive nested Dirichlet convolution: A063524, A008683 or A007427. - _Enrique Pérez Herrero_, Jul 12 2010
%Y A007428 Cf. A124010.
%K A007428 sign,easy,nice,mult
%O A007428 1,2
%A A007428 _N. J. A. Sloane_