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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=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<

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