A351319 a(n) = floor(n/k), where k is the nearest square to n.

1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 1

Author

Joelle H. Kassir, Mar 18 2022

Keywords

• nonn
• easy

Comments

For all n != 2, a(n) is 0 when less than the nearest square, A053187(n), and is 1 otherwise.

From Jon E. Schoenfield, Mar 22 2022: (Start)

After the first two terms, the sequence consists of runs of 0's and 1's, with run lengths 1,3,2,4,3,5,4,6,5,7,6,8,... = A028242.

For m >= 1, there are 2m integers k whose nearest square is m^2, namely, the m-1 integers (in the interval [m^2-m+1, m^2-1]) for which k < m^2 (hence a(k) = 0), followed by the m+1 integers (in the interval [m^2, m^2+m]) for which k >= m^2 (hence a(k) = 1). (End)

Examples

			a(5) = floor(5/4) = 1.
		

Crossrefs

Cf. A000194, A053187 (nearest square), A028242 (run lengths).

Cf. A267708 (essentially the same).

Programs

Mathematica

Table[Floor[n/Round[Sqrt[n]]^2], {n, 100}] (* Wesley Ivan Hurt, Mar 18 2022 *)

PARI

a(n) = if(n==2,2, my(r,s=sqrtint(n,&r)); r<=s); \\ Kevin Ryde, Mar 23 2022

Python

import math
def a(n):
    k = math.isqrt(n)
    if n - k**2 > k: k += 1
    return n // k**2;
for n in range(1, 101):
    print("{}, ".format(a(n)), end="")

Python

from math import isqrt
def A351319(n): return n if n <= 2 else int((k:=isqrt(n))**2+k-n+1 > 0) # Chai Wah Wu, Mar 26 2022

Formula

a(n) = floor(n/k), where k = round(sqrt(n))^2 = A053187(n).

a(n) = A267708(n) for n != 2.