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%I M3231 #61 Sep 13 2020 13:03:32
%S 1,4,4,10,4,16,4,20,10,16,4,40,4,16,16,35,4,40,4,40,16,16,4,80,10,16,
%T 20,40,4,64,4,56,16,16,16,100,4,16,16,80,4,64,4,40,40,16,4,140,10,40,
%U 16,40,4,80,16,80,16,16,4,160,4,16,40,84,16,64,4,40,16,64,4,200,4,16,40,40,16
%N d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.
%C Inverse Möbius transform applied thrice to all 1's sequence; or, Dirichlet convolution of d(n) (A000005).
%C Let n = Product p_i^e_i. tau (A000005) is tau_2, A007425 is tau_3, this sequence 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 Appears to equal the number of solid partitions of n that can be extended in exactly 4 ways to a solid partition of n + 1 by adding one element. - _Wouter Meeussen_, Sep 11 2004
%C Equals row sums of A127172. - _Gary W. Adamson_, Nov 05 2007
%D A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
%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="/A007426/b007426.txt">Table of n, a(n) for n = 1..10000</a>
%H O. Bordellès, <a href="http://www.emis.de/journals/JIPAM/article190.html">Explicit upper bounds for the average order of dn (m) and application to class number</a>, J. Inequal. Pure and Appl. Math, 3(3), 2002.
%H Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, Tomaž Pisanski, <a href="https://arxiv.org/abs/1911.02858">Regular Antilattices</a>, arXiv:1911.02858 [math.RA], 2019.
%H J. Furuya, Y. Tanigawa, W. Zhai, <a href="http://dx.doi.org/10.1007/s00605-010-0195-y">Dirichlet series obtained from the error term in the Dirichlet divisor problem</a>, Monatshefte für Mathematik, 2010, 160(4), 385-402.
%H J. Sándor, <a href="http://www.emis.de/journals/PM/53f1/9.html">On the arithmetical functions d~ k (n) and d^*~ k (n)</a>, Portugaliae Mathematica, 53, 107-116.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{d dividing n} tau(d)*tau(n/d). - _Benoit Cloitre_, May 12 2003
%F Dirichlet g.f.: zeta^4(x).
%F G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%p A007426 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(3+e[j][2],3), j=1..nops(e)); end;
%t tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 4], {n, 77}] (* _Robert G. Wilson v_, Nov 02 2005 *)
%t a[n_] := DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #]&]; Array[a, 80] (* _Jean-François Alcover_, Dec 01 2015 *)
%t tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 4], {n, 1, 100}] (* _Amiram Eldar_, Sep 13 2020 *)
%o (PARI) for(n=1,100,print1(sumdiv(n,k,sumdiv(k,x,numdiv(x))),","))
%o (PARI) a(n)=sumdiv(n,d,numdiv(n/d)*numdiv(d))
%o (PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+3, 3)) \\ _Charles R Greathouse IV_, Oct 28 2017
%Y Cf. A007425.
%Y Cf. A127172, A051731.
%Y Column k=4 of A077592.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Nov 02 2005
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