|
|
A007426
|
|
d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.
(Formerly M3231)
|
|
44
|
|
|
1, 4, 4, 10, 4, 16, 4, 20, 10, 16, 4, 40, 4, 16, 16, 35, 4, 40, 4, 40, 16, 16, 4, 80, 10, 16, 20, 40, 4, 64, 4, 56, 16, 16, 16, 100, 4, 16, 16, 80, 4, 64, 4, 40, 40, 16, 4, 140, 10, 40, 16, 40, 4, 80, 16, 80, 16, 16, 4, 160, 4, 16, 40, 84, 16, 64, 4, 40, 16, 64, 4, 200, 4, 16, 40, 40, 16
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Inverse Möbius transform applied thrice to all 1's sequence; or, Dirichlet convolution of d(n) (A000005).
Let n = Product p_i^e_i. tau (A000005) is tau_2, A007425 is tau_3, this sequence is tau_4, where tau_k(n) (also written as d_k(n)) = Product_i binomial(k-1+e_i, k-1) is the k-th Piltz function. It gives the number of ordered factorizations of n as a product of k terms.
Appears to equal the number of solid partitions of n that can be extended in exactly 4 ways to a solid partition of n + 1 by adding one element. - Wouter Meeussen, Sep 11 2004
|
|
REFERENCES
|
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.
|
|
FORMULA
|
a(n) = Sum_{d dividing n} tau(d)*tau(n/d). - Benoit Cloitre, May 12 2003
Dirichlet g.f.: zeta^4(x).
|
|
MAPLE
|
A007426 := proc(n) local e, j; e := ifactors(n)[2]: product(binomial(3+e[j][2], 3), j=1..nops(e)); end;
|
|
MATHEMATICA
|
tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 4], {n, 77}] (* Robert G. Wilson v, Nov 02 2005 *)
a[n_] := DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 4], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)
|
|
PROG
|
(PARI) for(n=1, 100, print1(sumdiv(n, k, sumdiv(k, x, numdiv(x))), ", "))
(PARI) a(n)=sumdiv(n, d, numdiv(n/d)*numdiv(d))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,mult
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|