A183300 Positive integers not of the form 2n^2.

1, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Author

Clark Kimberling, Jan 03 2011

Keywords

• nonn
• easy

Comments

Complement of A001105.

Integers whose number of even divisors (A183063) is even (for a proof, see A001105, the complement of this sequence), hence odd numbers (A005408) are a subsequence. - Bernard Schott, Sep 15 2021

Examples

			10 is in the sequence since 2*2^2=8 < 10 < 2*3^2=18.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005408, A183063.

Cf. A001105 (number of even divisors is odd), A028982 (number of odd divisors is odd), A028983 (number of odd divisors is even), this sequence (number of even divisors is even).

Programs

Magma

[n: n in [0..100] | not IsSquare(n/2)]; // Bruno Berselli, Dec 17 2013

Maple

A183300:=n->if type(sqrt(2*n)/2, integer) then NULL; else n; fi; seq(A183300(n), n=1..100); # Wesley Ivan Hurt, Dec 17 2013

Mathematica

a = 2; b = 0;
F[n_] := a*n^2 + b*n;
R[n_] := (n/a + ((b - 1)/(2a))^2)^(1/2);
G[n_] := n - 1 + Ceiling[R[n] - (b - 1)/(2a)];
Table[F[n], {n, 60}]
Table[G[n], {n, 100}] (* Clark Kimberling *)
r[n_] := Reduce[n == 2*k^2, k, Integers]; Select[Range[100], r[#] === False &] (* Jean-François Alcover, Dec 17 2013 *)
max = 100; Complement[Range[max], 2 Range[Ceiling[Sqrt[max/2]]]^2] (* Alonso del Arte, Dec 17 2013 *)
Module[{nn=10,f},Complement[Range[2nn^2],2Range[nn]^2]] (* Harvey P. Dale, Sep 06 2023 *)

PARI

is(n)=!issquare(n/2) \\ Charles R Greathouse IV, Sep 02 2015

PARI

a(n)=my(k=sqrtint(n\2)+n); if(k-sqrtint(k\2)Charles R Greathouse IV, Sep 02 2015

Python

from math import isqrt
def A183300(n): return n+(m:=isqrt(n>>1))+(n>(m+1)*((m<<1)+1)) # Chai Wah Wu, Aug 04 2025

Formula

a(n) = n + floor(sqrt(n/2) + 1/4). - Ridouane Oudra, Jan 26 2023

a(n) = n+m+1 if n>(m+1)*(2m+1) and a(n) = n+m otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Aug 04 2025

Extensions

Name clarified by Wesley Ivan Hurt, Dec 17 2013