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

1,2

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

Links

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

Formula

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

Example

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

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 *)

Prog

(Magma) [n: n in [0..100] | not IsSquare(n/2)]; // Bruno Berselli, Dec 17 2013
(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)<n, k+1, k) \\ Charles R Greathouse IV, Sep 02 2015

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).

Sequence in context: A354774 A194276 A184336 * A155854 A047252 A039235

Adjacent sequences: A183297 A183298 A183299 * A183301 A183302 A183303

Keyword

nonn,easy

Author

Clark Kimberling, Jan 03 2011

Extensions

Name clarified by Wesley Ivan Hurt, Dec 17 2013

Status

approved