Search: id:A007426
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%I A007426 M3231
%S A007426 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,
%T A007426 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,
%U A007426 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
%N A007426 d_4(n), or tau_4(n), the number of ordered factorizations of n as n = 
               rstu.
%C A007426 Inverse Moebius transform applied thrice to all 1's sequence; or, Dirichlet 
               convolution of d(n) [ A000005 ].
%C A007426 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.
%C A007426 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
%C A007426 Equals row sums of A127172. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 05 2007
%D A007426 A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
%D A007426 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A007426 T. D. Noe, <a href="b007426.txt">Table of n, a(n) for n=1..10000</a>
%H A007426 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A007426 a(n)=sum(d dividing n, tau(d)*tau(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 12 2003
%F A007426 Dirichlet g.f.: zeta^4(x)
%p A007426 A007426 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(3+e[j][2],
               3), j=1..nops(e)); end;
%t A007426 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 
               *)
%o A007426 (PARI) for(n=1,100,print1(sumdiv(n,k,sumdiv(k,x,numdiv(x))),","))
%o A007426 (PARI) a(n)=sumdiv(n,d,numdiv(n/d)*numdiv(d))
%Y A007426 Cf. A007425.
%Y A007426 Cf. A127172, A051731.
%Y A007426 Sequence in context: A091016 A120395 A160723 this_sequence A050348 A134637 
               A078910
%Y A007426 Adjacent sequences: A007423 A007424 A007425 this_sequence A007427 A007428 
               A007429
%K A007426 nonn,easy,mult
%O A007426 1,2
%A A007426 N. J. A. Sloane (njas(AT)research.att.com).
%E A007426 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 02 2005

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