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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.
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
OFFSET
1,1
COMMENTS
Every positive integer lies in exactly one of the sequences A247423 and A247524.
LINKS
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.
Sequence in context: A080152 A136994 A137288 * A116443 A243181 A236206
KEYWORD
nonn,easy,base
AUTHOR
Clark Kimberling, Sep 19 2014
STATUS
approved