A247523 -id:A247523

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A247524

Numbers k such that d(r,k) != d(s,k), where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.

+10
3

2, 4, 8, 11, 13, 14, 17, 18, 22, 24, 26, 27, 30, 32, 33, 34, 41, 42, 43, 45, 46, 47, 48, 50, 55, 60, 61, 62, 63, 64, 69, 71, 74, 76, 79, 81, 83, 90, 92, 98, 99, 100, 103, 105, 108, 109, 111, 112, 115, 117, 118, 123, 125, 126, 127, 132, 133, 137, 138, 143

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Every positive integer lies in exactly one of the sequences A247423 and A247524.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..2000

EXAMPLE

r has binary digits 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...

s has binary digits 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, ...

so that a(1) = 2 and a(2) = 4.

MATHEMATICA

z = 400; r1 = GoldenRatio; r = FractionalPart[r1]; s = FractionalPart[r1/2];

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

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

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

Flatten[Position[t, 1]] (* A247523 *)

Flatten[Position[t, 0]] (* A247524 *)

Module[{nn=150, gr, g2}, gr=Rest[RealDigits[GoldenRatio, 2, nn+1][[1]]]; g2 = RealDigits[ GoldenRatio/2, 2, nn][[1]]; Position[Thread[{gr, g2}], _?(#[[1]] != #[[2]]&), 1, Heads->False]]//Flatten (* Harvey P. Dale, Jun 28 2021 *)

CROSSREFS

Cf. A247523.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Sep 19 2014

STATUS

approved

A247635

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(8)}, and { } = fractional part.

+10
2

2, 8, 9, 10, 11, 14, 16, 17, 18, 20, 22, 24, 26, 28, 30, 31, 32, 33, 34, 35, 37, 39, 40, 43, 44, 45, 47, 49, 51, 54, 57, 58, 59, 62, 63, 67, 69, 73, 74, 75, 76, 79, 81, 82, 85, 87, 90, 92, 94, 97, 98, 106, 114, 115, 116, 117, 121, 123, 124, 125, 126, 128

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Every positive integer lies in exactly one of the sequences A247635 and A247636.

LINKS

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

EXAMPLE

r has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, ...

s has binary digits 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, ...

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

MATHEMATICA

z = 200; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[8]];

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

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

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

Flatten[Position[t, 1]] (* A247635 *)

Flatten[Position[t, 0]] (* A247636 *)