Problem 634

Problem 634

Numbers of the form $a^2b^3$

Define $F(n)$ to be the number of integers $x\le n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and both greater than $1$.

For example, $32=2^2\times 2^3$ and $72=3^2\times 2^3$ are the only two integers less than $100$ that can be written in this form. Hence, $F(100)=2$.

Further you are given $F(2\times 10^4)=130$ and $F(3\times 10^6)=2014$.

Find $F(9\times 10^{18})$.


考虑整数$x\le n$,若$x$可写成$x=a^2b^3$,其中$a$和$b$是大于$1$且可重复的整数,记所有此类整数的数目为$F(n)$。

例如,在小于$100$的整数中,只有$32=2^2\times 2^3$和$72=3^2\times 2^3$可以写成这种形式,因此$F(100)=2$。

此外,还已知$F(2\times 10^4)=130$以及$F(3\times 10^6)=2014$。

求$F(9\times 10^{18})$。