A111306 - OEIS

%I #32 Jun 12 2024 12:06:52

%S 1,12,12,78,12,144,12,364,78,144,12,936,12,144,144,1365,12,936,12,936,

%T 144,144,12,4368,78,144,364,936,12,1728,12,4368,144,144,144,6084,12,

%U 144,144,4368,12,1728,12,936,936,144,12,16380,78,936,144,936,12,4368,144

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

%H 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 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 G.f.: Sum_{k>=1} tau_11(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018

%F Multiplicative with a(p^e) = binomial(e+11,11). - _Amiram Eldar_, Sep 13 2020

%p b:= proc(n, k) option remember; `if`(k=1, 1,

%p       add(b(d, k-1), d=numtheory[divisors](n)))

%p     end:

%p a:= n-> b(n, 12):

%p seq(a(n), n=1..55);  # _Alois P. Heinz_, Jun 12 2024

%t 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 (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 (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 Cf. tau_2(n)...tau_11(n): A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221.

%Y Column k=12 of A077592.

%K mult,nonn

%O 1,2

%A _Gerald McGarvey_, Nov 02 2005