Problem 3

For any positive integer

n

, let

d(n)

denote the number of positive divisors of

n

(including

1

and

n

itself). Determine all positive integers

k

such that

\frac{d(n^2)}{d(n)}=k

for some

n

Step 3 of 6: Set up the converse

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In plain words

Build an exponent list from a product representation.

r_a=\frac{2a+1}{a+1}

Detailed analysis

It remains to show that every positive odd integer is a product of factors $r_a$ . If $k=1$ , use $a=0$ (the factor is $1$ ), corresponding to $n=1$ . We prove the representation by induction over odd $k$ .