%I #27 Sep 13 2020 13:04:32
%S 1,7,7,28,7,49,7,84,28,49,7,196,7,49,49,210,7,196,7,196,49,49,7,588,
%T 28,49,84,196,7,343,7,462,49,49,49,784,7,49,49,588,7,343,7,196,196,49,
%U 7,1470,28,196,49,196,7,588,49,588,49,49,7,1372,7,49,196,924,49,343,7,196
%N d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).
%H Seiichi Manyama, <a href="/A111217/b111217.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Enrique PĂ©rez Herrero)
%F Dirichlet convolution of A000012 with A034695, or of A000005 with A061200, or of A007425 with A007426. Dirichlet g.f. zeta^7(s). - _R. J. Mathar_, Apr 01 2011
%F G.f.: Sum_{k>=1} tau_6(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%F Multiplicative with a(p^e) = binomial(e+6,6). - _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, 7], {n, 68}] (* _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, 7], {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,x,numdiv(x)))))),","))
%o (PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+6, 6)) \\ _Charles R Greathouse IV_, Oct 28 2017
%Y Cf. tau_k(n) for k>=2: A000005, A007425, A007426, A061200, A034695, A111218 - A111221, A111306.
%Y Column k=7 of A077592.
%K mult,nonn
%O 1,2
%A _Gerald McGarvey_, Oct 25 2005