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

1, -7, -26, -7, -124, 182, -342, -7, -26, 868, -1330, 182, -2196, 2394, 3224, -7, -4912, 182, -6858, 868, 8892, 9310, -12166, 182, -124, 15372, -26, 2394, -24388, -22568, -29790, -7, 34580, 34384, 42408, 182, -50652, 48006, 57096, 868, -68920, -62244, -79506, 9310, 3224, 85162, -103822, 182

Offset: 1

Author

Vladeta Jovovic, Jul 26 2001

Keywords

• mult
• sign

Comments

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).

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

Dirichlet inverse of A059376. - R. J. Mathar, Jul 15 2010

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Crossrefs

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A046970(r = 2), A189922 (r = 4).

Cf. A027748.

Programs

Haskell

a063453 = product . map ((1 -) . (^ 3)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012

Maple

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:
A063453 := proc(n) Jinvk(n,3) ; end proc: # R. J. Mathar, Jul 04 2011
# second Maple program:
a:= n-> mul(1-i[1]^3, i=ifactors(n)[2]):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 26 2024

Mathematica

a[n_] := Total[MoebiusMu[#]*#^3& /@ Divisors[n]]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Jul 26 2011 *)
f[p_, e_] := (1-p^3); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

PARI

a(n) = sumdiv(n, d, moebius(d) * d^3); \\ Indranil Ghosh, Mar 11 2017

Python

from math import prod
from sympy import primefactors
def A063453(n): return prod(1-p**3 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

Formula

a(n) = Sum_{d|n} mu(d)*d^3.

Dirichlet g.f.: zeta(s)/zeta(s-3).

A023900(n) | a(n). - R. J. Mathar, Mar 30 2011

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.

G.f.: Sum_{k>=1} mu(k)*k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017

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