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Positive integers not of the form 2n^2.
4

%I #51 Sep 06 2023 16:21:06

%S 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,

%T 29,30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,53,

%U 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

%N Positive integers not of the form 2n^2.

%C Complement of A001105.

%C 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

%H Bruno Berselli, <a href="/A183300/b183300.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = n + floor(sqrt(n/2) + 1/4). - _Ridouane Oudra_, Jan 26 2023

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

%p 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

%t a = 2; b = 0;

%t F[n_] := a*n^2 + b*n;

%t R[n_] := (n/a + ((b - 1)/(2a))^2)^(1/2);

%t G[n_] := n - 1 + Ceiling[R[n] - (b - 1)/(2a)];

%t Table[F[n], {n, 60}]

%t Table[G[n], {n, 100}] (* _Clark Kimberling_ *)

%t r[n_] := Reduce[n == 2*k^2, k, Integers]; Select[Range[100], r[#] === False &] (* _Jean-François Alcover_, Dec 17 2013 *)

%t max = 100; Complement[Range[max], 2 Range[Ceiling[Sqrt[max/2]]]^2] (* _Alonso del Arte_, Dec 17 2013 *)

%t Module[{nn=10,f},Complement[Range[2nn^2],2Range[nn]^2]] (* _Harvey P. Dale_, Sep 06 2023 *)

%o (Magma) [n: n in [0..100] | not IsSquare(n/2)]; // _Bruno Berselli_, Dec 17 2013

%o (PARI) is(n)=!issquare(n/2) \\ _Charles R Greathouse IV_, Sep 02 2015

%o (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

%Y Cf. A005408, A183063.

%Y 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).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 03 2011

%E Name clarified by _Wesley Ivan Hurt_, Dec 17 2013