non prime factor

[man111]

how many non prime factors in the number [tex]N= 2^5 \times 3^7\times 9^2\times 11^4\times 13^3[/tex]

[rashi101]

[tex]1435[/tex]?

Solution:
[tex]N= 2^5 \times 3^7\times 9^2\times 11^4\times 13^3= 2^5 \times 3^{11}\times 11^4\times 13^3[/tex]

Each factor of N is in the form [tex]2^{a} \times 3^{b}\times 11^{c}\times 13^{d}[/tex], where [tex]a \in {0,1,2,3,4,5}[/tex], [tex]d\in0,1,2,3[/tex] etc. All possible combinations are [tex]6\times12\times5\times4=1440[/tex]. However, this also counts [tex]1, 2, 3, 11[/tex] and [tex]13[/tex], so the number of non prime factors is [tex]1440-5=1435[/tex].

I'd love to see a formal solution of this one - I've never been good at writing solutions .

[1089]

1 is not prime...

[rashi101]

Ops, right 1436, then?

[Станислав]

Obvious generalisation - the number of non prime factors of [tex]n=p^{\alpha_1}_1\ldots p^{\alpha_k}_k[/tex] is [tex]\tau(n)-k=(\alpha_1+1)\ldots(\alpha_k+1)-k[/tex].