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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
KEYWORD
sign,easy,nice,mult
AUTHOR
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)