A160086 a(n) = A104725(n) - A074206(n).

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, 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, 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

Offset

0,13

Comments

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)

Links

Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the b-files of A074206 and A104725)

A. O. Munagi, Labeled factorization of integers, Electron. J. Combin. 16 (2009), no. 1, Research Paper 50, 17pp.

Formula

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.

Example

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.

Maple

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);

Crossrefs

Cf. A074206, A104725.

Sequence in context: A299163 A326128 A369931 * A115869 A115859 A241663

Adjacent sequences: A160083 A160084 A160085 * A160087 A160088 A160089

Keyword

nonn

Author

Augustine O. Munagi, May 01 2009

Status

approved