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A007426
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d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.
(Formerly M3231)
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38
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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, 20, 40, 4, 64, 4, 56, 16, 16, 16, 100, 4, 16, 16, 80, 4, 64, 4, 40, 40, 16, 4, 140, 10, 40, 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
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OFFSET
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1,2
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COMMENTS
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Inverse Möbius transform applied thrice to all 1's sequence; or, Dirichlet convolution of d(n) (A000005).
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.
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
Equals row sums of A127172. - Gary W. Adamson, Nov 05 2007
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REFERENCES
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A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
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
O. Bordellès, Explicit upper bounds for the average order of dn (m) and application to class number, J. Inequal. Pure and Appl. Math, 3(3), 2002.
Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.
J. Furuya, Y. Tanigawa, W. Zhai, Dirichlet series obtained from the error term in the Dirichlet divisor problem, Monatshefte für Mathematik, 2010, 160(4), 385-402.
J. Sándor, On the arithmetical functions d~ k (n) and d^*~ k (n), Portugaliae Mathematica, 53, 107-116.
N. J. A. Sloane, Transforms
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FORMULA
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a(n) = Sum_{d dividing n} tau(d)*tau(n/d). - Benoit Cloitre, May 12 2003
Dirichlet g.f.: zeta^4(x).
G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018
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MAPLE
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A007426 := proc(n) local e, j; e := ifactors(n)[2]: product(binomial(3+e[j][2], 3), j=1..nops(e)); end;
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MATHEMATICA
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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 *)
a[n_] := DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)
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 *)
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PROG
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(PARI) for(n=1, 100, print1(sumdiv(n, k, sumdiv(k, x, numdiv(x))), ", "))
(PARI) a(n)=sumdiv(n, d, numdiv(n/d)*numdiv(d))
(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
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CROSSREFS
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Cf. A007425.
Cf. A127172, A051731.
Column k=4 of A077592.
Sequence in context: A160723 A255486 A286779 * A339336 A319056 A050348
Adjacent sequences: A007423 A007424 A007425 * A007427 A007428 A007429
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KEYWORD
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nonn,easy,mult
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Robert G. Wilson v, Nov 02 2005
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STATUS
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approved
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