

A007426


d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.
(Formerly M3231)


24



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

1,2


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(k1+e_i, k1) is the kth 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


REFERENCES

A. Ivic, The Riemann ZetaFunction, Wiley, NY, 1985, see p. xv.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n=1..10000
N. J. A. Sloane, Transforms


FORMULA

a(n)=sum(d dividing n, tau(d)*tau(n/d))  Benoit Cloitre, May 12 2003
Dirichlet g.f.: zeta^4(x)


MAPLE

A007426 := proc(n) local e, j; e := ifactors(n)[2]: product(binomial(3+e[j][2], 3), j=1..nops(e)); end;


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 *)


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))


CROSSREFS

Cf. A007425.
Cf. A127172, A051731.
Sequence in context: A205549 A120395 A160723 * A050348 A134637 A078910
Adjacent sequences: A007423 A007424 A007425 * A007427 A007428 A007429


KEYWORD

nonn,easy,mult


AUTHOR

N. J. A. Sloane.


EXTENSIONS

More terms from Robert G. Wilson v, Nov 02 2005


STATUS

approved



