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A183300
Positive integers not of the form 2n^2.
4
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
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. 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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 03 2011
EXTENSIONS
Name clarified by Wesley Ivan Hurt, Dec 17 2013
STATUS
approved