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A007427 Moebius transform applied twice to sequence 1,0,0,0,....
(Formerly M0198)
44
1, -2, -2, 1, -2, 4, -2, 0, 1, 4, -2, -2, -2, 4, 4, 0, -2, -2, -2, -2, 4, 4, -2, 0, 1, 4, 0, -2, -2, -8, -2, 0, 4, 4, 4, 1, -2, 4, 4, 0, -2, -8, -2, -2, -2, 4, -2, 0, 1, -2, 4, -2, -2, 0, 4, 0, 4, 4, -2, 4, -2, 4, -2, 0, 4, -8, -2, -2, 4, -8, -2, 0, -2, 4, -2, -2, 4, -8, -2, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

|a(n)| is the number of ways to write n as a product of 2 squarefree numbers (i.e., number of ways to write n = x*y with 1 <= x <= n, 1 <= y <= n, x and y squarefree). - Benoit Cloitre, Jan 01 2003

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.

Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.

Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

N. J. A. Sloane, Transforms.

Wikipedia, Adolf Piltz.

FORMULA

Dirichlet g.f.: 1/zeta(s)^2.

Multiplicative function with a(p^e) = binomial(2, e)*(-1)^e for p prime and e >= 0.

a(n) = Sum_{d|n} mu(d)*mu(n/d). - Benoit Cloitre, Apr 05 2002

a(n^2) = A008683(n)^2. a(A005117(n)) = (-2)^A001221(A005117(n)). - Enrique Pérez Herrero, Jun 27 2011 [Misrendering of contribution rectified by Peter Munn, Mar 06 2020]

a(n) is the Dirichlet inverse of A000005, which means a(n) = -Sum_{d|n, d<n} A000005(n/d)*a(d). - Enrique Pérez Herrero, Jan 19 2013

a(n) = 0 if n is not cubefree: A046099, otherwise sign(a(n))= lambda(n), where lambda is A008836. - Enrique Pérez Herrero, Jan 19 2013

Dirichlet g.f. of |a(n)|: zeta(s)^2/zeta(2s)^2 (conjectured). - Ralf Stephan, Jul 05 2013

a(n) = Sum_{k=1..A000005(n)} A225817(n,k)*A225817(n,n+1-k). - Reinhard Zumkeller, Jul 30 2013

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} tau(k)*A(x^k), where tau = A000005. - Ilya Gutkovskiy, May 11 2019

EXAMPLE

G.f. = x - 2*x^2 - 2*x^3 + x^4 - 2*x^5 + 4*x^6 - 2*x^7 + x^9 + 4*x^10 + ...

We have a(3^1) = C(2, 1)*(-1)^1 = -2, a(3^2) = C(2, 2)*(-1)^2 = 1, and a(3^m) = C(2, m)*(-1)^m = 0 for m >= 3. - Petros Hadjicostas, Jun 07 2019

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@@2)([1, seq(0, i=1..80)]); # Peter Luschny, Sep 08 2017

MATHEMATICA

f[n_] := Plus @@ Times @@@ (MoebiusMu[{#, n/#}] & /@ Divisors@n); Array[f, 105] (* Robert G. Wilson v *)

a[n_] := DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, (1 - X)^2)[n])}; /* Michael Somos, Nov 15 2002 */

(PARI) {a(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d)))}; /* Michael Somos, Nov 15 2002 */

(PARI) a(n)=if(n>1, my(f=factor(n)[, 2], s=sum(i=1, #f, f[i]==1)); if(vecmax(f)>2, 0, (-1)^s<<s), 1) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

a007427 n = sum $ zipWith (*) mds $ reverse mds where

   mds = a225817_row n

-- Reinhard Zumkeller, Jul 30 2013

CROSSREFS

Dirichlet inverse of A000005, Mobius transform of A008683.

Cf. A063524, A007428, A005117.

Sequence in context: A127677 A238009 A231145 * A048106 A304649 A228441

Adjacent sequences:  A007424 A007425 A007426 * A007428 A007429 A007430

KEYWORD

sign,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)