1,2

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.

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

A085782 gives the range of values of this sequence. - Matthew Vandermast (ghodges14(AT)comcast.net), 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 (lbeedassy(AT)hotmail.com), 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. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 14 2006

Equals row sums of triangle A143354 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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

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

N. J. A. Sloane, Transforms

a(n)=sum(d dividing n, tau(d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 04 2002

G.f.: sum(k>=1, tau(k)*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003

For n=Product p_i^e_i, a(n)=Product_i T(e_i + 1), where T(k)=k*(k+1)/2=A000217(k). - Lekraj Beedassy (lbeedassy(AT)hotmail.com), Sep 07 2004

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

Equals row sums of A127170. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 20 2007

Row sums of triangle A127170. Equals A134577 * [1/1, 1/2, 1/3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007

Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), 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 this sequence.

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

Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), 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^2)*tau_2(n)

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

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

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;

f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ a[n], {n, 90}] (from Robert G. Wilson v Sep 13 2004)

Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 08 2009: (Start)

tau[1, n_] := 1; SetAttributes[tau, Listable];

tau[k_, n_] := Plus @@ (tau[k - 1, Divisors[n]]) /; k > 1; (End)

(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]) (from R. Stephan)

Cf. A000005, A007426, A061201 (partial sums).

Cf. A127270.

Cf. A143354 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008]

Sequence in context: A110634 A151787 A113397 this_sequence A130695 A058587 A112163

Adjacent sequences: A007422 A007423 A007424 this_sequence A007426 A007427 A007428

nonn,nice,easy,mult,new

N. J. A. Sloane (njas(AT)research.att.com).

Maple program and comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 13 2004