Sequences Involving Binary & Prime-Factorization

[Leroy Quet]

Anyone know the asymptotics of {a(j)} and/or {b(j)}??




>
>First, we have the sequence formed by simply letting
>
>a(m) = 1, if the sum of all exponents of the prime-factorization of m has 
>no carries when summed in in base-2.
>
>a(m) = 0, if there are any carries in the summing of the exponents of the 
>prime-factorization of m.
>

>...

>a(k) -> 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1,...
>
>(Defining a(1) = 1 makes b()-recursion below work.)
>
>
>Now, let b(m) = 
>
>sum{k=1 to m} a(k).
>
>b(m) gives the number of positive integers  <= m and with an a() of 1, 
>obviously.

>....

 [recursive definition of b(m) snipped]

>...

>
>b(k) -> 1, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 11, 12, 13,...
>

thanks,
Leroy Quet