%I #65 Jul 28 2022 09:07:10

%S 3,5,7,8,10,11,13,14,15,17,18,19,21,22,23,24,26,27,28,29,31,32,33,34,

%T 35,37,38,39,40,41,43,44,45,46,47,48,50,51,52,53,54,55,57,58,59,60,61,

%U 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,82,83,84,85,86,87

%N Complement of quarter-squares (A002620).

%C Intersection of A000037 and A078358. - _Reinhard Zumkeller_, May 08 2012

%C Numbers k such that floor(sqrt(k)+1/2) does not divide k. - _Wesley Ivan Hurt_, Dec 01 2020

%H Reinhard Zumkeller, <a href="/A049068/b049068.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F A237347(a(n)) = 2. - _Reinhard Zumkeller_, Mar 18 2014

%F A240025(a(n)) = 0. - _Reinhard Zumkeller_, Jul 05 2014

%F a(n) = A080037(n) - 1. - _Peter Kagey_, Dec 08 2015

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

%F = (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

%F a(n) = A000037(A000037(n)). - _Gerald Hillier_, Dec 01 2017

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

%t max = 100; Complement[Range[0, max], Table[Quotient[n^2, 4], {n, 0, 2*Sqrt[max]}]] (* _Jean-François Alcover_, Apr 18 2013 *)

%t Table[n + Ceiling[2 * Sqrt[n]], {n, 100}] (* _Wesley Ivan Hurt_, Mar 01 2014 *)

%o (PARI) {a(n) = if( n<1, 0, n+1 + sqrtint(4*n - 3))} /* _Michael Somos_, Oct 16 2006 */

%o (Haskell)

%o a049068 n = a049068_list !! (n-1)

%o a049068 = filter ((== 0) . a240025) [0..]

%o -- _Reinhard Zumkeller_, Jul 05 2014, Mar 18 2014, May 08 2012

%o (Magma) [n+Ceiling(2*Sqrt(n)): n in [1..70]]; // _Vincenzo Librandi_, Dec 09 2015

%o (Python)

%o from math import isqrt

%o def A049068(n): return n+1+isqrt((n<<2)-1) # _Chai Wah Wu_, Jul 27 2022

%Y Complement: A002620.

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

%K nonn,nice,easy

%O 1,1

%A _Michael Somos_, Aug 06 1999