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A247455 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 = {3*sqrt(2)}, and { } = fractional part. 6
1, 8, 9, 10, 11, 15, 21, 25, 29, 38, 42, 48, 51, 54, 57, 58, 59, 62, 64, 66, 70, 72, 78, 81, 82, 86, 89, 93, 96, 107, 109, 111, 113, 122, 128, 130, 134, 136, 139, 144, 147, 148, 149, 151, 153, 161, 162, 165, 169, 173, 181, 182, 183, 187, 191, 195, 200, 202 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Every positive integer lies in exactly one of these: A247455, A247456, A247457, A247758.  Let s denote any of these; what can be said about lim(#s < n)/n, 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

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

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

so that a(1) = 2 and a(2) = 8.

MATHEMATICA

z = 400; r = FractionalPart[Sqrt[2]]; s = FractionalPart[3*Sqrt[2]];

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}];

t3 = Table[If[u[[n]] == 1 && v[[n]] == 0, 1, 0], {n, 1, z}];

t4 = Table[If[u[[n]] == 1 && v[[n]] == 1, 1, 0], {n, 1, z}];

Flatten[Position[t1, 1]]  (* A247455 *)

Flatten[Position[t2, 1]]  (* A247456 *)

Flatten[Position[t3, 1]]  (* A247457 *)

Flatten[Position[t4, 1]]  (* A247458 *)

CROSSREFS

Cf. A246356, A247456, A247457, A247458.

Sequence in context: A297260 A296701 A297134 * A280290 A138581 A323062

Adjacent sequences:  A247452 A247453 A247454 * A247456 A247457 A247458

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Sep 18 2014

STATUS

approved

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Last modified December 8 01:46 EST 2019. Contains 329850 sequences. (Running on oeis4.)