A183299 - OEIS

%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