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 6 of 6: Realize every representation

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In plain words

Turn the factor representation into an integer.

k=\prod_i\frac{2a_i+1}{a_i+1}\Rightarrow n=\prod_i p_i^{a_i}

Detailed analysis

Choose distinct primes $p_i$ and set $n=\prod p_i^{a_i}$ (omitting any exponent $a_i=0$ if desired). The divisor formula then realizes the chosen product. Thus every positive odd $k$ occurs, and the answer is precisely all positive odd integers.