Divisors [III]

Divisors [III]

The blurb for this problem is the same as Divisors [I], save for the statement of the problem.

Consider the sequence of all natural numbers, and then for each number, listing each of the divisors of this number in increasing order:

becomes:

Since this (infinite) sequence has a lot of duplicates, we can notate for each digit whether it is the first, second, third... occurence using a subscript:

For this third problem, we are considering taking the reciprocal of every value in this sequence, and summing it:

For this problem, we want you to estimate the value of the partial sum up to and including some $i_1$ (The first occurence of some value $i$)

Input

Input will contain a single value i, representing the sequence value $i_1$

Output

Output an estimate of the partial sum up to and including the first occurence of $\frac{1}{i}$.

In order to receive $30\%$ of the grade, your solution should have maximum error at most $\ln(i)$.

In order to be judged as correct and receive full marks, your solution should have maximum error at most $\frac{1}{i} + 2\sqrt{i}\ln ( 1 + \frac{\ln{i}}{i} ) + \sqrt{i}*2^{-55+\log_2(i)}$. For reference, this error looks like the following (note the logarithmic axis):

Constraints

• $1 \leq i \leq 10^{11}$

Example

5

6.783333333333333

Although anything $\pm 1.447997$ from this answer would suffice.