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a(n) = A104725(n) - A074206(n).
1

%I #9 May 25 2017 09:36:54

%S 0,0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,7,0,3,0,3,0,0,0,25,0,0,1,3,0,6,0,36,

%T 0,0,0,36,0,0,0,25,0,6,0,3,3,0,0,152,0,3,0,3,0,25,0,25,0,0,0,69,0,0,3,

%U 171,0,6,0,3,0,6,0,279,0,0,3,3,0,6,0,152,7,0,0,69,0,0,0,25,0,69,0,3,0,0,0

%N a(n) = A104725(n) - A074206(n).

%C a(n) is also the excess of the number of labeled factorizations of n over the number of ordered factorizations (see the Munagi link for definition of labeled factorization)

%H Antti Karttunen, <a href="/A160086/b160086.txt">Table of n, a(n) for n = 0..10000</a> (computed from the b-files of A074206 and A104725)

%H A. O. Munagi, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r50">Labeled factorization of integers</a>, Electron. J. Combin. 16 (2009), no. 1, Research Paper 50, 17pp.

%F a(n) = Sum(ordfac(n,k)*(Bell(k-1)-1),k=1..Omega(n)), where ordfac(n,k)=number of ordered factorizations of n into k factors.

%e a(8)=1 because A074206(8)=4 and A104725(8)=5, so a(8)=5-4. The only labeled factorization of 8 which is not an ordered factorization is (2_1.2_3)(2_2). a(9)=0 because A074206(9)=2=A104725(9). The labeled factorizations of 9, namely (9_1) and (3_1)(3_2), are also ordered factorizations.

%p a:=proc(n::integer) local u, r, i, j, k; if n<2 then return 0; end if; u:=map(x->x[2], ifactors(n)[2]); r:=add(u[i], i=1..nops(u)); add(add((-1)^i*binomial(k, i)*product(binomial(u[j]+k-i-1, u[j]), j=1..nops(u)), i=0..k-1)*(bell(k-1)-1), k=1..r); end proc: seq(a(n),n=0..99);

%Y Cf. A074206, A104725.

%K nonn

%O 0,13

%A _Augustine O. Munagi_, May 01 2009