A007426 d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

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

Crossrefs

Cf. A007425.

Cf. A127172, A051731, A061202 (partial sums).

Column k=4 of A077592.

Programs

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 *)

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

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^4)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

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

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

Extensions

More terms from Robert G. Wilson v, Nov 02 2005