A030140 The nonsquares squared.

4, 9, 25, 36, 49, 64, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The complement of the fourth powers A000583 within the squares A000290. - Peter Munn, Aug 20 2019

Examples

			a(1)=2^2, a(2)=3^2, a(3)=5^2, a(4)=6^2, a(5)=7^2, ..., a(n)=(integer which is not a perfect square)^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000037, A000290, A000583, A013661, A013662, A062503.

Positions of 2's in A352080.

Related to A016945 via A225546.

Programs

Magma

[(n + Floor(1/2 + Sqrt(n)))^2: n in [1..60]]; // Vincenzo Librandi, Apr 06 2020

Maple

a:=proc(n) if type(sqrt(n),integer)=false then n^2 else fi end: seq(a(n),n=1..70); # Emeric Deutsch, Apr 11 2007

Mathematica

a[n_] := (n + Floor[1/2 + Sqrt[n]])^2;
Array[a, 50] (* Jean-François Alcover, Apr 05 2020 *)

Python

from math import isqrt
def A030140(n): return (n+(k:=isqrt(n))+int(n>=k*(k+1)+1))**2 # Chai Wah Wu, Jun 17 2024

Formula

a(n) = A000037(n)^2.

Sum_{n>=1} 1/a(n) = zeta(2) - zeta(4) = A013661 - A013662 = 0.5626108331... - Amiram Eldar, Nov 14 2020

{a(n) : n >= 1} = {A225546(6m+3) : m >= 0}. - Peter Munn, Nov 17 2022

Extensions

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar