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A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.
(Formerly M2282)
69
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

N. J. A. Sloane, Transforms

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

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 a squarefree number (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)

EXAMPLE

a(6) = 9; The divisors of 6 are {1,2,3,6} and the number 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 *)

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, A007426, A061201 (partial sums), A127270, A143354, A027750.

Cf. A175596.

Sequence in context: A151787 A113397 A226602 * A260152 A130695 A181788

Adjacent sequences:  A007422 A007423 A007424 * A007426 A007427 A007428

KEYWORD

nonn,nice,easy,mult

AUTHOR

N. J. A. Sloane, May 24 1994

STATUS

approved

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Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.