MathDB

Define a positive integer

n

to be squarish if either

n

is itself a perfect square or the distance from

n

to the nearest perfect square is a perfect square. For example,

2016

is squarish, because the nearest perfect square to

2016

45^2=2025

and

2025-2016=9

is a perfect square. (Of the positive integers between

1

and

10,

only

6

and

7

are not squarish.)
For a positive integer

N,

let

S(N)

be the number of squarish integers between

1

and

N,

inclusive. Find positive constants

\alpha

and

\beta

such that

\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,

or show that no such constants exist.

Problem Statement