A111221 - OEIS

%I #28 Sep 13 2020 13:05:21

%S 1,11,11,66,11,121,11,286,66,121,11,726,11,121,121,1001,11,726,11,726,

%T 121,121,11,3146,66,121,286,726,11,1331,11,3003,121,121,121,4356,11,

%U 121,121,3146,11,1331,11,726,726,121,11,11011,66,726,121,726,11,3146,121

%N d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).

%H Seiichi Manyama, <a href="/A111221/b111221.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_10(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018

%F Multiplicative with a(p^e) = binomial(e+10,10). - _Amiram Eldar_, Sep 13 2020

%t tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 11], {n, 55}] (* _Robert G. Wilson v_, Nov 02 2005 *)

%t tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 11], {n, 1, 100}] (* _Amiram Eldar_, Sep 13 2020 *)

%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,x,numdiv(x)))))))))),","))

%o (PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+10, 10)) \\ _Charles R Greathouse IV_, Oct 28 2017

%Y Cf. tau_1(n): A000012

%Y Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.

%Y Cf. tau_7(n)...tau_10(n): A111217, A111218, A111219, A111220.

%Y Cf. tau_12(n): A111306.

%Y Column k=11 of A077592.

%K mult,nonn

%O 1,2

%A _Gerald McGarvey_, Oct 25 2005