A049068 Complement of quarter-squares (A002620).

3, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87

Offset: 1

Author

Michael Somos, Aug 06 1999

Keywords

• nonn
• nice
• easy

Comments

Intersection of A000037 and A078358. - Reinhard Zumkeller, May 08 2012

Numbers k such that floor(sqrt(k)+1/2) does not divide k. - Wesley Ivan Hurt, Dec 01 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Complement: A002620.

Cf. A000037, A027434, A078358, A080037, A237347, A240025.

Programs

Haskell

a049068 n = a049068_list !! (n-1)
a049068 = filter ((== 0) . a240025) [0..]
-- Reinhard Zumkeller, Jul 05 2014, Mar 18 2014, May 08 2012

Magma

[n+Ceiling(2*Sqrt(n)): n in [1..70]]; // Vincenzo Librandi, Dec 09 2015

Maple

A049068:=n->n + ceil(2*sqrt(n)); seq(A049068(n), n=1..100); # Wesley Ivan Hurt, Mar 01 2014

Mathematica

max = 100; Complement[Range[0, max], Table[Quotient[n^2, 4], {n, 0, 2*Sqrt[max]}]]  (* Jean-François Alcover, Apr 18 2013 *)
Table[n + Ceiling[2 * Sqrt[n]], {n, 100}] (* Wesley Ivan Hurt, Mar 01 2014 *)
Select[Range[100],Mod[#,Floor[Sqrt[#]+1/2]]!=0&] (* Harvey P. Dale, May 27 2025 *)

PARI

{a(n) = if( n<1, 0, n+1 + sqrtint(4*n - 3))} /* Michael Somos, Oct 16 2006 */

Python

from math import isqrt
def A049068(n): return n+1+isqrt((n<<2)-1) # Chai Wah Wu, Jul 27 2022

Formula

a(n) = n + A027434(n).

Other identities and observations. For all n >= 1:

A237347(a(n)) = 2. - Reinhard Zumkeller, Mar 18 2014

A240025(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2014

a(n) = A080037(n) - 1. - Peter Kagey, Dec 08 2015

G.f.: x/(1-x)^2 + Sum_{k>=0} (x^(1+k^2)*(1+x^k))/(1-x)

= (x*Theta3(x)+ x^(3/4)*Theta2(x))/(2-2*x) + (3-x)*x/(2*(1-x)^2) where Theta3 and Theta2 are Jacobi Theta functions. - Robert Israel, Dec 09 2015

a(n) = A000037(A000037(n)). - Gerald Hillier, Dec 01 2017