%I #28 Feb 04 2022 14:42:56
%S 1,2,4,5,6,7,9,10,11,12,13,14,16,17,18,19,20,21,22,23,25,26,27,28,29,
%T 30,31,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,54,
%U 55,56,57,58,59,60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,81,82,83,84,85,86,87,88
%N Complement of A005563.
%C From _Andres Cicuttin_, Apr 18 2016: (Start)
%C Defining the probability for integer value x in a Poisson distribution of integer mean = q > 0 as P(q,x) = e^(-q)*(q^x)/x! then it is conjectured that P(n, a(n)+1)- P(n, a(n)) <= P(n, k+1)- P(n, k), for every k > 0.
%C That is, a(n) is the position where the Poisson distribution with mean n has its minimum discrete difference (not proved, but tested up to n = 20*10^3).
%C (Very qualitative) Plot of a Poisson Distribution with mean q = n. The vertical line above a(n) indicates the place where the distribution has its minimum (negative) discrete difference.
%C P
%C ^
%C | *
%C | * *
%C | * *
%C | * *
%C | * *
%C | * | *
%C | * | *
%C | * | *
%C | * | *
%C *-------------+----+----------------------> x
%C n a(n)
%C For instance, if n = 8 then
%C P(8, a(8)+1) - P(8, a(8)) = P(8,11) - P(8,10) = -0.027071
%C If we now calculate the discrete difference in a(n)+1 we then obtain
%C P(8,a(8)+2) - P(8,a(8)+1) = P(8,12)- P (8,11) = -0.0240634
%C and in a(n)-1
%C P(8,a(8)) - P(8,a(8)-1) = P(8,10) - P(8,9) = -0.0248154
%C Both previous values are larger than the minimum obtained at a(n). (End)
%C Numbers k such that sqrt(k+1) is not an integer. - _Wesley Ivan Hurt_, Feb 03 2022
%F (See the Mathematica code.)
%t a=1; b=2;
%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}]
%Y Cf. A005563.
%K nonn
%O 1,2
%A _Clark Kimberling_, Jan 03 2011