codeforces#P1891D. Suspicious logarithms
Suspicious logarithms
Description
Let $f$($x$) be the floor of the binary logarithm of $x$. In other words, $f$($x$) is largest non-negative integer $y$, such that $2^y$ does not exceed $x$.
Let $g$($x$) be the floor of the logarithm of $x$ with base $f$($x$). In other words, $g$($x$) is the largest non-negative integer $z$, such that ${f(x)}^{z}$ does not exceed $x$.
You are given $q$ queries. The $i$-th query consists of two integers $l_i$ and $r_i$. The answer to the query is the sum of $g$($k$) across all integers $k$, such that $l_i \leq k \leq r_i$. Since the answers might be large, print them modulo ${10^9 + 7}$.
The first line contains a single integer $q$ — the number of queries ($1 \leq q \leq 10^5$).
The next $q$ lines each contain two integers $l_i$ and $r_i$ — the bounds of the $i$-th query ($4 \leq l_i \leq r_i \leq 10^{18}$).
For each query, output the answer to the query modulo $10^9 + 7$.
Input
The first line contains a single integer $q$ — the number of queries ($1 \leq q \leq 10^5$).
The next $q$ lines each contain two integers $l_i$ and $r_i$ — the bounds of the $i$-th query ($4 \leq l_i \leq r_i \leq 10^{18}$).
Output
For each query, output the answer to the query modulo $10^9 + 7$.
12
4 6
4 7
4 8
4 100000
179 1000000000000000000
57 179
4 201018959
7 201018960
729 50624
728 50624
728 50625
729 50625
6
8
9
348641
41949982
246
1
0
149688
149690
149694
149692
Note
The table below contains the values of the functions $f$($x$) and $g$($x$) for all $x$ such that $1 \leq x \leq 8$.
| $x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| $f$ | $0$ | $1$ | $1$ | $2$ | $2$ | $2$ | $2$ | $3$ |
| $g$ | $-$ | $-$ | $-$ | $2$ | $2$ | $2$ | $2$ | $1$ |