|
| |
|
|
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
|
|
|
|
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.
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]):
|
|
|
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
|
|
|
|
KEYWORD
|
sign,easy,mult
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
