%I #19 Sep 13 2020 13:04:43
%S 1,8,8,36,8,64,8,120,36,64,8,288,8,64,64,330,8,288,8,288,64,64,8,960,
%T 36,64,120,288,8,512,8,792,64,64,64,1296,8,64,64,960,8,512,8,288,288,
%U 64,8,2640,36,288,64,288,8,960,64,960,64,64,8,2304,8,64,288,1716,64,512,8
%N d_8(n), tau_8(n), number of ordered factorizations of n as n = rstuvwxy (8-factorizations).
%H Vincenzo Librandi, <a href="/A111218/b111218.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: Sum_{k>=1} tau_7(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%F Multiplicative with a(p^e) = binomial(e+7,7). - _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, 8], {n, 67}] (* _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, 8], {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,x,numdiv(x))))))),","))
%o (PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+7, 7)) \\ _Charles R Greathouse IV_, Oct 28 2017
%Y Column k=8 of A077592.
%K mult,nonn
%O 1,2
%A _Gerald McGarvey_, Oct 25 2005