

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 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(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
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), 385402.
J. Sándor, On the arithmetical functions d~ k (n) and d^*~ k (n), Portugaliae Mathematica, 53, 107116.
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, Nov 02 2005 *)
a[n_] := DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #]&]; Array[a, 80] (* JeanFrançois Alcover, Dec 01 2015 *)


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))
(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


CROSSREFS

Cf. A007425.
Cf. A127172, A051731.
Column k=4 of A077592.
Sequence in context: A160723 A255486 A286779 * 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



