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A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

%I A111306 #32 Jun 12 2024 12:06:52
%S A111306 1,12,12,78,12,144,12,364,78,144,12,936,12,144,144,1365,12,936,12,936,
%T A111306 144,144,12,4368,78,144,364,936,12,1728,12,4368,144,144,144,6084,12,
%U A111306 144,144,4368,12,1728,12,936,936,144,12,16380,78,936,144,936,12,4368,144
%N A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).
%H A111306 Seiichi Manyama, <a href="/A111306/b111306.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Enrique Pérez Herrero)
%H A111306 Adolf Piltz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN271032898">Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst</a>, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
%F A111306 G.f.: Sum_{k>=1} tau_11(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%F A111306 Multiplicative with a(p^e) = binomial(e+11,11). - _Amiram Eldar_, Sep 13 2020
%p A111306 b:= proc(n, k) option remember; `if`(k=1, 1,
%p A111306       add(b(d, k-1), d=numtheory[divisors](n)))
%p A111306     end:
%p A111306 a:= n-> b(n, 12):
%p A111306 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jun 12 2024
%t A111306 tau[k_,1]:=1; tau[k_,n_]:=Times@@(Binomial[#+k-1,k-1]&/@FactorInteger[n][[All,2]]); Table[tau[12,n],{n,1000}] (* _Enrique Pérez Herrero_, Jan 17 2013 *)
%o A111306 (PARI) for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,m,sumdiv(m,o,sumdiv(o,p,sumdiv(p,q,sumdiv(q,r,sumdiv(r,x,numdiv(x))))))))))),","))
%o A111306 (PARI) a(n,f=factor(n))=f=f[,2]; prod(i=1,#f, binomial(f[i]+11, 11)) \\ _Charles R Greathouse IV_, Oct 28 2017
%Y A111306 Cf. tau_2(n)...tau_11(n): A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221.
%Y A111306 Column k=12 of A077592.
%K A111306 mult,nonn
%O A111306 1,2
%A A111306 _Gerald McGarvey_, Nov 02 2005