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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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25
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Inverse Moebius 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 (qntmpkt(AT)yahoo.com), Nov 05 2007
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REFERENCES
| 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
| T. D. Noe, Table of n, a(n) for n=1..10000
N. J. A. Sloane, Transforms
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FORMULA
| a(n)=sum(d dividing n, tau(d)*tau(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 12 2003
Dirichlet g.f.: zeta^4(x)
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MAPLE
| 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
| 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 *)
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PROG
| (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))
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CROSSREFS
| Cf. A007425.
Cf. A127172, A051731.
Sequence in context: A205549 A120395 A160723 * A050348 A134637 A078910
Adjacent sequences: A007423 A007424 A007425 * A007427 A007428 A007429
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KEYWORD
| nonn,easy,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 02 2005
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