While playing around with programming, a problem suddenly came across my mind.
Given $n$ let's say $2^{64}$ (64-bit unsigned integer). Find a positive integer $x$, $1 \leq x \leq n$, such that $x$ has maximum divisors.
My attempt was:
Let $n = p_1^{a_1} \times p_2^{a_2} \times ... \times p_k^{a_k}.$
The number of divisors is given by: $(a_1 + 1) \times (a_2 + 1) \times ... \times (a_k + 1)$
We need to find the maximum of these products. Since $(a_1 + 1)_{max} \implies {a_1}_{max}$, the product becomes: $a_1 \times a_2 \times .... \times a_k$
Let $P = a_1 \times a_2 \times .... \times a_k$.
The maximum of $P$ means $a_i = a_j$ where $1 \leq i, j \leq k$.
So I guess my question is how can we find a number that is less than $n$ which has this property?
Thanks,