n=b(1)*b(2)*..(b(1)+b(2)+..)

[Leroy Quet]

Let a(n) = the number of ways that
n = (product{k=1 to m} b(k)) *(sum{k=1 to m} b(k)),
where the b's are positive integers. m and the b's may be different for 
different ways to get to n.

For example,
6 = 1*1*1*1*1*1*(1+1+1+1+1+1) =
1*2*(1+2) {and = 2*1*(2+1)}.

So, there are really two sequences I am wondering about: the number of 
{b(k)}'s,  based on n, where different orderings of the same b's are 
counted separately, and the number of{b(k)}'s where different orderings 
of the same b's are not considered distinct.

So, under the former definition of the sequence, a(6) = 3
(because of 1*2*(1+2) and 2*1*(2+1) are being counted separately).
Under the latter definition, a(6) = 2.

I get, and I may very well be wrong,
the sequence (where different orderings of the same b's are not 
considered distinct} beginning:
1,2,1,2,1,2,1,2,2,2,1,3,...

Are either of the two sequences defined above already in the EIS 
(probably under a different name)?

Could someone please calculate/submit either sequence not already in the 
EIS?

thanks,
Leroy Quet