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A246356

Numbers k such that d(r,k) = 0 and d(s,k) = 0, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

6, 9, 12, 20, 24, 28, 29, 37, 48, 52, 57, 58, 62, 66, 69, 81, 82, 89, 93, 96, 102, 104, 106, 111, 113, 122, 129, 130, 139, 144, 149, 151, 159, 161, 163, 165, 166, 177, 179, 181, 186, 187, 190, 191, 195, 201, 202, 204, 217, 219, 220, 222, 225, 228, 232, 233

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OFFSET

1,1

COMMENTS

Every positive integer lies in exactly one of these: A246356, A246357, A246358, A247356. Let s denote any of these; is lim(#s < n)/n = 1/4, where (#s < n) represents the number of numbers in s that are < n?

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

EXAMPLE

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

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

so that a(1) = 6.

MATHEMATICA

z = 500; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];

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

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

t1 = Table[If[u[[n]] == 0 && v[[n]] == 0, 1, 0], {n, 1, z}];

t2 = Table[If[u[[n]] == 0 && v[[n]] == 1, 1, 0], {n, 1, z}];