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%I M2282 #146 Oct 30 2023 09:35:32
%S 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,
%T 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,
%U 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
%N d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.
%C 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_
%C Inverse Möbius transform applied twice to all 1's sequence.
%C A085782 gives the range of values of this sequence. - _Matthew Vandermast_, Jul 12 2004
%C 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 Number of divisors of n's divisors. - _Lekraj Beedassy_, Sep 07 2004
%C 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
%C Equals row sums of A127170. - _Gary W. Adamson_, May 20 2007
%C Equals A134577 * [1/1, 1/2, 1/3, ...]. - _Gary W. Adamson_, Nov 02 2007
%C Equals row sums of triangle A143354. - _Gary W. Adamson_, Aug 10 2008
%C 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
%C Also row sums of A195050. - _Omar E. Pol_, Nov 26 2015
%C 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
%C Number of ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - _Wesley Ivan Hurt_, Mar 22 2022
%D M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
%D A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
%D Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007425/b007425.txt">Table of n, a(n) for n = 1..10000</a>
%H R. Eisinga, R. Breitling, and T. Heskes, <a href="http://dx.doi.org/10.1016/j.febslet.2013.01.037">The exact probability distribution of the rank product statistics for replicated experiments</a>, FEBS Letters, 2013, 587: 677-682, doi:10.1016/j.febslet.2013.01.037
%H Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, and Tomaž Pisanski, <a href="https://arxiv.org/abs/1911.02858">Regular Antilattices</a>, arXiv:1911.02858 [math.RA], 2019.
%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.
%H Adolf Piltz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN271032898">Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst</a>, 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.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H Qingfeng Sun and Deyu Zhang, <a href="http://arxiv.org/abs/1510.06170">Sums of the triple divisor function over values of a ternary quadratic form</a>, arXiv:1510.06170 [math.NT], 2015.
%H E. C. Titchmarsh, <a href="https://doi.org/10.1093/qmath/os-13.1.129">Some problems in the analytic theory of numbers</a>, The Quarterly Journal of Mathematics 1 (1942): 129-152.
%H Wikipedia, <a href="https://de.wikipedia.org/wiki/Adolf_Piltz">Adolf Piltz</a>.
%F a(n) = Sum_{d dividing n} tau(d). - _Benoit Cloitre_, Apr 04 2002
%F G.f.: Sum_{k>=1} tau(k)*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F For n = Product p_i^e_i, a(n) = Product_i A000217(e_i + 1). - _Lekraj Beedassy_, Sep 07 2004
%F Dirichlet g.f.: zeta^3(s).
%F From _Enrique Pérez Herrero_, Nov 03 2009: (Start)
%F 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.
%F a(s) = 3^omega(s), if s>1 is squarefree (A005117) and omega(s) is: A001221. (End)
%F From _Enrique Pérez Herrero_, Nov 08 2009: (Start)
%F 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 tau_3(n) >= 2*tau_2(n) - 1.
%F tau_3(n) <= tau_2(n)^2 + tau_2(n)-1. (End)
%F From _Vladimir Shevelev_, Dec 22 2017: (Start)
%F a(n) = sqrt(Sum_{d|n}(tau(d))^3);
%F a(n) = |Sum_{d|n} A008836(d)*(tau(d))^2)|.
%F 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)
%F 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
%e 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.
%e 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
%p 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 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(2+e[j][2],2), j=1..nops(e)); end; # _Len Smiley_
%t f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ f[n], {n, 90}] (* _Robert G. Wilson v_, Sep 13 2004 *)
%t 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 *)
%t Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 50}] (* _Wesley Ivan Hurt_, May 30 2014 *)
%t 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 *)
%o (PARI) for(n=1,100,print1(sumdiv(n,k,numdiv(k)),","))
%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3)[n]) \\ _Ralf Stephan_
%o (PARI) a(n)=sumdiv(n, x, sumdiv(x, y, 1 )) \\ _Joerg Arndt_, Oct 07 2012
%o (PARI) a(n)=sumdivmult(n,k,numdiv(k)) \\ _Charles R Greathouse IV_, Aug 30 2013
%o (Haskell)
%o a007425 = sum . map a000005 . a027750_row
%o -- _Reinhard Zumkeller_, Feb 16 2012
%Y Cf. A000005 (Mobius transform), A007426 (inverse Mobius transform), A061201 (partial sums), A127270, A143354, A027750, A007428 (Dirichlet inverse), A175596.
%Y Column k=3 of A077592.
%Y Additional cross-references mentioned in a comment: A034836, A038548, A140733.
%K nonn,nice,easy,mult
%O 1,2
%A _N. J. A. Sloane_, May 24 1994
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