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A189922 Jordan function J_{-4} multiplied by n^4. 8
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, -923520, -15, 1171200, 1252800, 1497600, 1200, -1874160 (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
a(n) = Sum_{d divides n} d * sigma_3(d)^(-1) * sigma_1(n/d), where sigma_3(n)^(-1) = A053825(n) denotes the Dirichlet inverse of sigma_3(n). - Peter Bala, Jan 26 2024
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.
MAPLE
a:= n-> mul(1-i[1]^4, i=ifactors(n)[2]):
seq(a(n), n=1..48); # Alois P. Heinz, Jan 26 2024
MATHEMATICA
a[n_] := Sum[ MoebiusMu[d]*d^4, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := (1-p^4); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)
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: A082540 A372952 A269657 * A085808 A180577 A033594
KEYWORD
sign,easy,mult
AUTHOR
Wolfdieter Lang, Jun 16 2011
STATUS
approved

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Last modified May 22 13:18 EDT 2024. Contains 372755 sequences. (Running on oeis4.)