Number-Theoretical Fibonacci-Like Sequence

[Leroy Quet]

Posted this to sci.math:

---

I may have made a mistake in calculating terms, but this sequence does
not yet seem to be in the EIS.

Let a(1) = a(2) = 1;

For m >= 2,
let a(m+1) = a(m) + a(j(m)),

where j(m) is the largest PROPER divisor of m.

(I have probably over-represented this sequence by calling it
"Fibonacci-like".)

The sequence begins (?):

1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 18, 23,...

Is there a closed-form for the a(m)'s? 

Thanks,
Leroy Quet

PS: There are probably other good examples of sequences combining somehow 
number-theory and Fibonacci numbers.