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

[Max]

Leroy Quet wrote:
> 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, 

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 4, 10, 1, 13, 1, 15, 6, 11, 1, 27, 2, 13, 8, 28, 1, 27, 1, 36, 10, 17, 4, 62, 1, 19, 12, 59, 1, 47, 1, 66, 19, 23, 1, 118, 2, 31, 16, 91, 1, 78, 8, 117, 18, 29, 1, 193, 1, 31, 26, 159, 10, 115, 1, 153, 22, 51, 1, 320, 1, 37, 35, 190, 6, 161, 1, 296, 37, 41, 1, 387, 14, 43, 28, 355, 1, 258, 8, 276, 30, 47, 16, 662, 1, 57, 56, 362

> and the number of{b(k)}'s where different orderings
> of the same b's are not considered distinct.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 4, 3, 2, 1, 9, 2, 3, 2, 4, 1, 6, 2, 7, 2, 2, 1, 8, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 11, 1, 2, 3, 4, 2, 5, 1, 9, 4, 2, 1, 10, 2, 2, 2, 7, 1, 9, 2, 4, 2, 2, 2, 13, 1, 3, 4, 7

I've user the following PARI program to compute first 100 terms of these sequences.

{f(p, s) = for(a=1,100,if(p*a*(s+a)>100,break,c[p*a*(s+a)]++;f(p*a,s+a)))}
{a1() = c=vector(100);f(1,0);c}

{g(p, s, b) = for(a=b,100,if(p*a*(s+a)>100,break,c[p*a*(s+a)]++;g(p*a,s+a,a)))}
{a2() = c=vector(100);g(1,0,1);c}

> 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,...

We agree except a(2). Why do you have a(2)=2?

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

Neither of them is in OEIS.

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

Go ahead. You're the author. ;)

Max