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A063453 Multiplicative with a(p^e) = 1 - p^3. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

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

MATHEMATICA

a[n_] := Total[MoebiusMu[#]*#^3& /@ Divisors[n]]; Table[a[n], {n, 1, 48}] (* Jean-Fran├žois Alcover, Jul 26 2011 *)

PROG

(Haskell)

a063453 = product . map ((1 -) . (^ 3)) . a027748_row

-- Reinhard Zumkeller, Jan 19 2012

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

CROSSREFS

Cf. A023900, A046970.

Cf. A027748.

Sequence in context: A214593 A012490 A157702 * A284054 A284786 A262109

Adjacent sequences:  A063450 A063451 A063452 * A063454 A063455 A063456

KEYWORD

mult,sign

AUTHOR

Vladeta Jovovic, Jul 26 2001

STATUS

approved

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Last modified September 25 16:31 EDT 2017. Contains 292499 sequences.