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A049068 Complement of quarter-squares (A002620). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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
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 *)
PROG
(PARI) {a(n) = if( n<1, 0, n+1 + sqrtint(4*n - 3))} /* Michael Somos, Oct 16 2006 */
(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
(Python)
from math import isqrt
def A049068(n): return n+1+isqrt((n<<2)-1) # Chai Wah Wu, Jul 27 2022
CROSSREFS
Complement: A002620.
Sequence in context: A175146 A351076 A171508 * A185602 A225554 A257019
KEYWORD
nonn,nice,easy
AUTHOR
Michael Somos, Aug 06 1999
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)