%I A007425 M2282
%S A007425 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,
%T A007425 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,
%U A007425 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
%N A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = 
               rst.
%C A007425 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.
%C A007425 Inverse Moebius transform applied twice to all 1's sequence.
%C A007425 A085782 gives the range of values of this sequence. - Matthew Vandermast 
               (ghodges14(AT)comcast.net), Jul 12 2004
%C A007425 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
%C A007425 Number of divisors of n's divisors. - Lekraj Beedassy (lbeedassy(AT)hotmail.com), 
               Sep 07 2004
%C A007425 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
%C A007425 Equals row sums of triangle A143354 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 10 2008]
%D A007425 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007425 M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; 
               p. 239.
%D A007425 A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
%D A007425 Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.
%H A007425 T. D. Noe, <a href="b007425.txt">Table of n, a(n) for n=1..10000</a>
%H A007425 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Sequences realized as Parker vectors ...</
               a>, J. Integer Seqs., Vol. 6, 2003.
%H A007425 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A007425 a(n)=sum(d dividing n, tau(d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 04 2002
%F A007425 G.f.: sum(k>=1, tau(k)*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 21 2003
%F A007425 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
%F A007425 Dirichlet g.f.: zeta^3(x)
%F A007425 Equals row sums of A127170. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 20 2007
%F A007425 Row sums of triangle A127170. Equals A134577 * [1/1, 1/2, 1/3,...]. - 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007
%F A007425 Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 03 
               2009: (Start)
%F A007425 a(n^2)=tau_3(n^2)=tau_2(n^2)*tau_2(n), where tau_2 is A000005 and tau_3 
               this sequence.
%F A007425 a(s)=3^omega(s), if s>1 is a squarefree number (A005117) and omega(s) 
               is: A001221 (End)
%F A007425 Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 08 
               2009: (Start)
%F A007425 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
%F A007425 tau_3(n^2)=tau_2(n^2)*tau_2(n)
%F A007425 tau_3(n)>=2*tau_2(n)-1
%F A007425 tau_3(n)<=tau_2(n)^2+tau_2(n)-1 (End)
%p A007425 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;
%p A007425 A007425 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(2+e[j][2],
               2), j=1..nops(e)); end;
%t A007425 f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ a[n], {n, 90}] 
               (from Robert G. Wilson v Sep 13 2004)
%t A007425 Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 08 
               2009: (Start)
%t A007425 tau[1, n_] := 1; SetAttributes[tau, Listable];
%t A007425 tau[k_, n_] := Plus @@ (tau[k - 1, Divisors[n]]) /; k > 1; (End)
%o A007425 (PARI) for(n=1,100,print1(sumdiv(n,k,numdiv(k)),","))
%o A007425 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3)[n]) (from R. Stephan)
%Y A007425 Cf. A000005, A007426, A061201 (partial sums).
%Y A007425 Cf. A127270.
%Y A007425 Cf. A143354 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008]
%Y A007425 Sequence in context: A110634 A151787 A113397 this_sequence A130695 A058587 
               A112163
%Y A007425 Adjacent sequences: A007422 A007423 A007424 this_sequence A007426 A007427 
               A007428
%K A007425 nonn,nice,easy,mult,new
%O A007425 1,2
%A A007425 N. J. A. Sloane (njas(AT)research.att.com).
%E A007425 Maple program and comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).
%E A007425 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 13 2004