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A007428
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Moebius transform applied thrice to sequence 1,0,0,0,....
(Formerly M2271)
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21
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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
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OFFSET
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1,2
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COMMENTS
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abs(a(n)) is the number of ways to write n=xyz where x,y,z are squarefree numbers. - Benoit Cloitre, Jan 02 2018
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (3 choose e) (-1)^e.
Dirichlet g.f.: 1/zeta(s)^3.
a(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End)
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MAPLE
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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:
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MATHEMATICA
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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;
a[n_] := Which[n==1, 1, PrimeQ[n], -3, True, Times @@ Map[Function[e, Binomial[3, e] (-1)^e], FactorInteger[n][[All, 2]]]];
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PROG
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(Haskell)
a007428 n = product
[a007318' 3 e * cycle [1, -1] !! fromIntegral e | e <- a124010_row n]
(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
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CROSSREFS
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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STATUS
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approved
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