A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t. (Formerly M2282)

1, 3, 3, 6, 3, 9, 3, 10, 6, 9, 3, 18, 3, 9, 9, 15, 3, 18, 3, 18, 9, 9, 3, 30, 6, 9, 10, 18, 3, 27, 3, 21, 9, 9, 9, 36, 3, 9, 9, 30, 3, 27, 3, 18, 18, 9, 3, 45, 6, 18, 9, 18, 3, 30, 9, 30, 9, 9, 3, 54, 3, 9, 18, 28, 9, 27, 3, 18, 9, 27, 3, 60, 3, 9, 18, 18, 9, 27, 3, 45, 15, 9, 3, 54, 9, 9, 9, 30, 3

Offset

1,2

Comments

Let n = Product p_i^e_i. Tau (A000005) is tau_2, this sequence is tau_3, A007426 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. - Len Smiley

Inverse Möbius transform applied twice to all 1's sequence.

A085782 gives the range of values of this sequence. - Matthew Vandermast, Jul 12 2004

Appears to equal the number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. - Wouter Meeussen, Sep 11 2004

Number of divisors of n's divisors. - Lekraj Beedassy, Sep 07 2004

Number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. If the partition is not a box, there is a minimal i+j where b_{i,j} != b_{1,1} and an element can be added there. - Franklin T. Adams-Watters, Jun 14 2006

Equals row sums of A127170. - Gary W. Adamson, May 20 2007

Equals A134577 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 02 2007

Equals row sums of triangle A143354. - Gary W. Adamson, Aug 10 2008

a(n) is congruent to 1 (mod 3) if n is a perfect cube, otherwise a(n) is congruent to 0 (mod 3). - Geoffrey Critzer, Mar 20 2015

Also row sums of A195050. - Omar E. Pol, Nov 26 2015

Number of 3D grids of n congruent boxes with three different edge lengths, in a box, modulo rotation (cf. A034836 for cubes instead of boxes and A140773 for boxes with two different edge lengths; cf. A000005 for the 2D case). - Manfred Boergens, Apr 06 2021

Number of ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

References

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.

A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.

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

R. Eisinga, R. Breitling, and T. Heskes, The exact probability distribution of the rank product statistics for replicated experiments, FEBS Letters, 2013, 587: 677-682, doi:10.1016/j.febslet.2013.01.037

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

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.

Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

N. J. A. Sloane, Transforms.

Qingfeng Sun and Deyu Zhang, Sums of the triple divisor function over values of a ternary quadratic form, arXiv:1510.06170 [math.NT], 2015.

E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.

Wikipedia, Adolf Piltz.

Formula

a(n) = Sum_{d dividing n} tau(d). - Benoit Cloitre, Apr 04 2002

G.f.: Sum_{k>=1} tau(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

For n = Product p_i^e_i, a(n) = Product_i A000217(e_i + 1). - Lekraj Beedassy, Sep 07 2004

Dirichlet g.f.: zeta^3(s).

From Enrique Pérez Herrero, Nov 03 2009: (Start)

a(n^2) = tau_3(n^2) = tau_2(n^2)*tau_2(n), where tau_2 is A000005 and tau_3 is this sequence.

a(s) = 3^omega(s), if s>1 is squarefree (A005117) and omega(s) is: A001221. (End)

From Enrique Pérez Herrero, Nov 08 2009: (Start)

a(n) = tau_3(n) = tau_2(n)*tau_2(n*rad(n))/tau_2(rad(n)), where rad(n) is A007947 and tau_2(n) is A000005.

tau_3(n) >= 2*tau_2(n) - 1.

tau_3(n) <= tau_2(n)^2 + tau_2(n)-1. (End)

From Vladimir Shevelev, Dec 22 2017: (Start)

a(n) = sqrt(Sum_{d|n}(tau(d))^3);

a(n) = |Sum_{d|n} A008836(d)*(tau(d))^2)|.

The first formula follows from the first Cloitre formula and a Liouville formula; the second formula follows from our analogous formula (cf. our comment in Formula section of A000005). (End)

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

Example

a(6) = 9; the divisors of 6 are {1,2,3,6} and the numbers of divisors of these divisors are 1, 2, 2, and 4. Adding them, we get 9 as a result.
Also, since 6 is a squarefree number, the formula from Herrero can be used to obtain the result: a(6) = 3^omega(6) = 3^2 = 9. - Wesley Ivan Hurt, May 30 2014

Maple

f:=proc(n) local t1, i, j, k; t1:=0; for i from 1 to n do for j from 1 to n do for k from 1 to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;
A007425 := proc(n) local e, j; e := ifactors(n)[2]: product(binomial(2+e[j][2], 2), j=1..nops(e)); end; # Len Smiley

Mathematica

f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Sep 13 2004 *)
SetAttributes[tau, Listable]; tau[1, n_] := 1; tau[k_, n_] := Plus @@ (tau[k-1, Divisors[n]]); Table[tau[3, n], {n, 100}] (* Enrique Pérez Herrero, Nov 08 2009 *)
Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 50}] (* Wesley Ivan Hurt, May 30 2014 *)
f[p_, e_] := (e+1)*(e+2)/2;  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 27 2019 *)

Prog

(PARI) for(n=1, 100, print1(sumdiv(n, k, numdiv(k)), ", "))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)^3)[n]) \\ Ralf Stephan
(PARI) a(n)=sumdiv(n, x, sumdiv(x, y, 1 )) \\ Joerg Arndt, Oct 07 2012
(PARI) a(n)=sumdivmult(n, k, numdiv(k)) \\ Charles R Greathouse IV, Aug 30 2013
(Haskell)
a007425 = sum . map a000005 . a027750_row
-- Reinhard Zumkeller, Feb 16 2012

Crossrefs

Cf. A000005 (Mobius transform), A007426 (inverse Mobius transform), A061201 (partial sums), A127270, A143354, A027750, A007428 (Dirichlet inverse), A175596.

Column k=3 of A077592.

Additional cross-references mentioned in a comment: A034836, A038548, A140733.

Sequence in context: A341343 A226602 A307000 * A260152 A358223 A130695

Adjacent sequences: A007422 A007423 A007424 * A007426 A007427 A007428

Keyword

nonn,nice,easy,mult

Author

N. J. A. Sloane, May 24 1994

Status

approved