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 A007428 Moebius transform applied thrice to sequence 1,0,0,0,.... (Formerly M2271) 21
 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, -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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Dirichlet inverse of A007425. - R. J. Mathar, Jul 15 2010 abs(a(n)) is the number of ways to write n=xyz where x,y,z are squarefree numbers. - Benoit Cloitre, Jan 02 2018 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000 N. J. A. Sloane, Transforms FORMULA Multiplicative with a(p^e) = (3 choose e) (-1)^e. Dirichlet g.f.: 1/zeta(s)^3. From Enrique Pérez Herrero, Jul 12 2010: (Start) a(n^3) = A008683(n). a(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End) a(A046101(n)) = 0. - Enrique Pérez Herrero, Sep 07 2017 a(n) = Sum_{a*b*c=n} mu(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022 MAPLE möbius := proc(a) local b, i, mo: b := NULL: mo := (m, n) -> `if`(irem(m, n) = 0, numtheory:-mobius(m/n), 0); for i to nops(a) do b := b, add(mo(i, j)*a[j], j=1..i) od: [b] end: (möbius@@3)([1, seq(0, i=1..77)]); # Peter Luschny, Sep 08 2017 MATHEMATICA tau[1, n_Integer]:=1; SetAttributes[tau, Listable]; tau[k_Integer, n_Integer]:=Plus@@(tau[k-1, Divisors[n]])/; k > 1; tau[k_Integer, n_Integer]:=Plus@@(tau[k+1, Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1; A007428[n_]:=tau[ -3, n]; (* Enrique Pérez Herrero, Jul 12 2010 *) a[n_] := Which[n==1, 1, PrimeQ[n], -3, True, Times @@ Map[Function[e, Binomial[3, e] (-1)^e], FactorInteger[n][[All, 2]]]]; Array[a, 100] (* Jean-François Alcover, Jun 20 2018 *) PROG (Haskell) a007428 n = product [a007318' 3 e * cycle [1, -1] !! fromIntegral e | e <- a124010_row n] -- Reinhard Zumkeller, Oct 09 2013 (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 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021 CROSSREFS Consecutive nested Dirichlet convolution: A063524, A008683 or A007427. - Enrique Pérez Herrero, Jul 12 2010 Cf. A124010. Sequence in context: A222292 A245441 A333793 * A184099 A074816 A203564 Adjacent sequences: A007425 A007426 A007427 * A007429 A007430 A007431 KEYWORD sign,easy,nice,mult AUTHOR STATUS approved

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Last modified December 5 09:12 EST 2022. Contains 358585 sequences. (Running on oeis4.)