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A007425
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d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.
(Formerly M2282)
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59
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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
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OFFSET
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1,2
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COMMENTS
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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, 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 triangle A143354 [From Gary W. Adamson, Aug 10 2008]
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REFERENCES
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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
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).
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LINKS
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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
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FORMULA
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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 T(e_i + 1), where T(k)=k*(k+1)/2=A000217(k). - Lekraj Beedassy, Sep 07 2004
Dirichlet g.f.: zeta^3(x)
Equals row sums of A127170. - Gary W. Adamson, May 20 2007
Row sums of triangle A127170. Equals A134577 * [1/1, 1/2, 1/3,...]. - Gary W. Adamson, 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)
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MAPLE
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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;
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MATHEMATICA
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f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ f[n], {n, 90}] (from 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}] (* Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 08 2009 *)
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PROG
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(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 Ralf Stephan */
(Haskell)
a007425 = sum . map a000005 . a027750_row
-- Reinhard Zumkeller, Feb 16 2012
(PARI) a(n)=sumdiv(n, x, sumdiv(x, y, 1 ) ); /* Joerg Arndt, Oct 07 2012 */
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CROSSREFS
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Cf. A000005, A007426, A061201 (partial sums).
Cf. A127270.
Cf. A143354 [From Gary W. Adamson, Aug 10 2008]
Cf. A027750.
Sequence in context: A110634 A151787 A113397 * A130695 A181788 A058587
Adjacent sequences: A007422 A007423 A007424 * A007426 A007427 A007428
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KEYWORD
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nonn,nice,easy,mult,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Maple program and comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).
More terms from Robert G. Wilson v, Sep 13 2004
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STATUS
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approved
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