A111220 - OEIS

%I #26 Sep 13 2020 13:05:03

%S 1,10,10,55,10,100,10,220,55,100,10,550,10,100,100,715,10,550,10,550,

%T 100,100,10,2200,55,100,220,550,10,1000,10,2002,100,100,100,3025,10,

%U 100,100,2200,10,1000,10,550,550,100,10,7150,55,550,100,550,10,2200,100

%N d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).

%H Seiichi Manyama, <a href="/A111220/b111220.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_9(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018

%F Multiplicative with a(p^e) = binomial(e+9,9). - _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, 10], {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, 10], {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,x,numdiv(x))))))))),","))

%o (PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+9, 9)) \\ _Charles R Greathouse IV_, Oct 28 2017

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

%Y Column k=10 of A077592.

%K mult,nonn

%O 1,2

%A _Gerald McGarvey_, Oct 25 2005