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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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31
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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
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 *)
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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 * A319056 A050348 A134637
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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