Question

Numbers

Answer 1

ok

Answer 2

How many positive integers \(b\) \(\left[\color{orangered}{b <1000} \right] \) are there such that \(\large a^{\lfloor a \rfloor}=b\) has a solution for \(a\).

Answer 3

this is college stuf man

Answer 4

\(\lfloor . \rfloor \) is the floor function

Answer 5

does this help

Answer 6

): sorry but all that is not related to the my question

Answer 7

Let's reverse-solve it.

Consider all integers in the interval\[n^n \le \cdots < (n+1)^n\]They'll satisfy your equation for sure.
Do \(n = 1\Rightarrow b = 1\)
\(n = 2\Rightarrow b \in [4,9)\)
\(n=3\Rightarrow b \in [27, 64)\)\[n=4\Rightarrow b \in [256, 625)\]We'll not consider \(n \ge 5\) because \(b \) cannot exceed thousand, and we're gonna be looking at numbers like \(5^5\) which do exceed thousand.

Answer 8

does this help

Answer 9

all these are google searches and this question is not available on google
so that won't help

Answer 10

As I highlighted above, the possible solutions for \(b\) are:\[\{1\} \cup [4, 9)\cup [27, 64) \cup [256, 625) \]The number of integers in this interval:\[= 1 + 5 + 37 + 369 = \boxed{412}\]

Answer 11

412

Answer 12

412 is correct! :D

Answer 13

do you wanna read a poem?

Answer 14

@imqwerty

Answer 15

is it 412