A247454 - OEIS

%I #11 Sep 26 2014 17:23:14

%S 3,5,6,7,9,12,16,17,19,20,22,23,24,28,29,30,32,33,37,41,45,48,49,52,

%T 56,57,58,61,62,66,67,69,74,75,76,81,82,88,89,90,91,93,96,98,99,101,

%U 102,104,105,106,108,111,113,115,116,117,120,122,125,129,130,131

%N Numbers k such that d(r,k) = d(s,k), where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

%C Every positive integer lies in exactly one of the sequences A247454 and A247324.  Let s denote either sequence; is lim(#s < n)/n = 1/2, where (#s < n) represents the number of numbers in s that are < n?

%H Clark Kimberling, <a href="/A247454/b247454.txt">Table of n, a(n) for n = 1..1000</a>

%e {sqrt(2)} has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,...

%e {sqrt(3)} has binary digits 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0,..

%e so that a(1) = 3 and a(2) = 5.

%t z = 200; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];

%t u = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[r, 2, z]];

%t v = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[s, 2, z]];

%t t = Table[If[u[[n]] == v[[n]], 1, 0], {n, 1, z}];

%t Flatten[Position[t, 1]]  (* A247454 *)

%t Flatten[Position[t, 0]]  (* A247324 *)

%Y Cf. A246356, A247324.

%K nonn,easy,base

%O 1,1

%A _Clark Kimberling_, Sep 17 2014