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A063453 Multiplicative with a(p^e) = 1 - p^3.

%I A063453 #54 Nov 11 2024 05:52:23
%S A063453 1,-7,-26,-7,-124,182,-342,-7,-26,868,-1330,182,-2196,2394,3224,-7,
%T A063453 -4912,182,-6858,868,8892,9310,-12166,182,-124,15372,-26,2394,-24388,
%U A063453 -22568,-29790,-7,34580,34384,42408,182,-50652,48006,57096,868,-68920,-62244,-79506,9310,3224,85162,-103822,182
%N A063453 Multiplicative with a(p^e) = 1 - p^3.
%C A063453 More generally, Dirichlet g.f. for Sum_{d|n} mu(d)*d^k, the Dirichlet inverse of the Jordan function J_k, is zeta(s)/zeta(s-k).
%C A063453 Apart from different signs also Sum_{d|n} core(d)^3*mu(n/d) where core(x) is the squarefree part of x. - _Benoit Cloitre_, May 31 2002
%C A063453 Dirichlet inverse of A059376. - _R. J. Mathar_, Jul 15 2010
%D A063453 T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.
%H A063453 Reinhard Zumkeller, <a href="/A063453/b063453.txt">Table of n, a(n) for n = 1..10000</a>
%H A063453 P. G. Brown, <a href="http://www.jstor.org/stable/3621931">Some comments on inverse arithmetic functions</a>, Math. Gaz. 89 (516) (2005) 403-408.
%F A063453 a(n) = Sum_{d|n} mu(d)*d^3.
%F A063453 Dirichlet g.f.: zeta(s)/zeta(s-3).
%F A063453 A023900(n) | a(n). - _R. J. Mathar_, Mar 30 2011
%F A063453 a(n)= product_{p|n}(1-p^3), n>=2, p prime, a(1)=1.  a(n)= J_{-3}(n)*n^3, with the Jordan function J_k(n). See the Apostol reference, p. 48, exercise 17. - _Wolfdieter Lang_, Jun 16 2011.
%F A063453 G.f.: Sum_{k>=1} mu(k)*k^3*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 15 2017
%F A063453 a(n) = Sum_{d divides n} d * sigma_2(d)^(-1) * sigma_1(n/d), where sigma_2(n)^(-1) = A053822(n) denotes the Dirichlet inverse of sigma_2(n). - _Peter Bala_, Jan 26 2024
%p A063453 Jinvk := proc(n,k) local a,f,p ; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; a := a*(1-p^k) ; end do: a ; end proc:
%p A063453 A063453 := proc(n) Jinvk(n,3) ; end proc: # _R. J. Mathar_, Jul 04 2011
%p A063453 # second Maple program:
%p A063453 a:= n-> mul(1-i[1]^3, i=ifactors(n)[2]):
%p A063453 seq(a(n), n=1..48);  # _Alois P. Heinz_, Jan 26 2024
%t A063453 a[n_] := Total[MoebiusMu[#]*#^3& /@ Divisors[n]]; Table[a[n], {n, 1, 48}] (* _Jean-François Alcover_, Jul 26 2011 *)
%t A063453 f[p_, e_] := (1-p^3); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 08 2020 *)
%o A063453 (Haskell)
%o A063453 a063453 = product . map ((1 -) . (^ 3)) . a027748_row
%o A063453 -- _Reinhard Zumkeller_, Jan 19 2012
%o A063453 (PARI) a(n) = sumdiv(n, d, moebius(d) * d^3); \\ _Indranil Ghosh_, Mar 11 2017
%o A063453 (Python)
%o A063453 from math import prod
%o A063453 from sympy import primefactors
%o A063453 def A063453(n): return prod(1-p**3 for p in primefactors(n)) # _Chai Wah Wu_, Sep 08 2023
%Y A063453 Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A046970(r = 2), A189922 (r = 4).
%Y A063453 Cf. A027748.
%K A063453 mult,sign
%O A063453 1,2
%A A063453 _Vladeta Jovovic_, Jul 26 2001