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A189922 Jordan function J_{-4} multiplied by n^4. 6
1, -15, -80, -15, -624, 1200, -2400, -15, -80, 9360, -14640, 1200, -28560, 36000, 49920, -15, -83520, 1200, -130320, 9360, 192000, 219600, -279840, 1200, -624, 428400, -80, 36000, -707280, -748800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the Jordan function J_k see the Comtet and Apostol references.

REFERENCES

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

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..200 from Indranil Ghosh)

FORMULA

a(n) = J_{-4}(n)*n^4 = Product_{p prime | n} (1 - p^4), for n>=2, a(1)=1.

a(n) = Sum_{d|n} mu(d)*d^4 with the Moebius function mu = A008683.

Dirichlet g.f.: zeta(s)/zeta(s-4).

Sum identity: Sum_{d|n} a(n)*(n/d)^4 = 1 for all n>=1.

a(n) = a(rad(n)) with rad(n) = A007947(n), the squarefree kernel of n.

G.f.: Sum_{k>=1} mu(k)*k^4*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017

EXAMPLE

a(2) = a(4) = a(8) = ... = 1 - 2^4 = -15.

a(4) = mu(1)*1^4 + mu(2)*2^4 + mu(4)*4^4 = 1 - 16 + 0 = -15.

Sum identity for n=4: a(1)*(4/1)^4 + a(2)*(4/2)^4 + a(4)*(4/4)^4 = 256 - 15*16 - 15 = 1.

MATHEMATICA

a[n_] := Sum[ MoebiusMu[d]*d^4, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 03 2012 *)

PROG

(PARI) for (n=1, 30, print1(sumdiv(n, d, moebius(d) * d^4), ", ")); \\ Indranil Ghosh, Mar 11 2017

CROSSREFS

Cf. A023900 (k=-1), A046970 (k=-2), A063453 (k=-3).

Sequence in context: A212741 A082540 A269657 * A085808 A180577 A033594

Adjacent sequences:  A189919 A189920 A189921 * A189923 A189924 A189925

KEYWORD

sign,easy,mult

AUTHOR

Wolfdieter Lang, Jun 16 2011

STATUS

approved

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Last modified August 8 11:12 EDT 2020. Contains 336293 sequences. (Running on oeis4.)