A007426 - OEIS

A007426

d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.
(Formerly M3231)

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

Equals row sums of A127172. - Gary W. Adamson, Nov 05 2007

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

T. D. Noe, Table of n, a(n) for n = 1..10000

O. Bordellès, Explicit upper bounds for the average order of dn (m) and application to class number, J. Inequal. Pure and Appl. Math, 3(3), 2002.

Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.

J. Furuya, Y. Tanigawa, W. Zhai, Dirichlet series obtained from the error term in the Dirichlet divisor problem, Monatshefte für Mathematik, 2010, 160(4), 385-402.

J. Sándor, On the arithmetical functions d~ k (n) and d^*~ k (n), Portugaliae Mathematica, 53, 107-116.

N. J. A. Sloane, Transforms

FORMULA

a(n) = Sum_{d dividing n} tau(d)*tau(n/d). - Benoit Cloitre, May 12 2003

Dirichlet g.f.: zeta^4(x).

G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

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))

(PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+3, 3)) \\ Charles R Greathouse IV, Oct 28 2017

(Python)

from math import prod, comb

from sympy import factorint

def A007426(n): return prod(comb(3+e, 3) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

CROSSREFS

Cf. A007425.

Cf. A127172, A051731.

Column k=4 of A077592.