A247423 -id:A247423 - cp's OEIS frontend

A247523 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.

1, 3, 5, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 23, 25, 28, 29, 31, 35, 36, 37, 38, 39, 40, 44, 49, 51, 52, 53, 54, 56, 57, 58, 59, 65, 66, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 101, 102, 104, 106, 107, 110

Offset: 1

Clark Kimberling, Sep 19 2014

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

			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) = 1 and a(2) = 3.

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

Cf. A247524, A247519.

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 *)

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.

Original entry on oeis.org

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

Clark Kimberling, Sep 19 2014

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

			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.

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

Cf. A247523.

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 *)

A247424 Odd numbers not of the form 2*A005206(A003249(m)) - 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 59, 63, 65, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 105, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named y' in [Carlitz] (cf. proof of Thm. 7.31), which defines it as the complement of A247423.

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

cp's OEIS Frontend

A247523 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.

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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.

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A247424 Odd numbers not of the form 2*A005206(A003249(m)) - 1.

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