A120750 -id:A120750 - cp's OEIS frontend

A120243 Numbers k such that {k*sqrt(2)} < 1/2, where { } = fractional part.

1, 3, 5, 6, 8, 10, 13, 15, 17, 18, 20, 22, 25, 27, 29, 30, 32, 34, 35, 37, 39, 42, 44, 46, 47, 49, 51, 54, 56, 58, 59, 61, 63, 66, 68, 71, 73, 75, 76, 78, 80, 83, 85, 87, 88, 90, 92, 95, 97, 99, 100, 102, 104, 105, 107, 109, 112, 114, 116, 117, 119, 121, 124, 126, 128, 129

Offset: 1

Clark Kimberling, Jul 01 2006

nonn

The complement of a is b=A120749. Is a(n) < b(n) for all n? If k is a positive integer, then is b(n) - a(n) = k for infinitely many n?

			{r} = {1.4142...} = 0.4142... < 1/2, so a(1)=1.
{2r} = 0.828... > 1/2, so b(1) = 2, where b = complement of a.
{3r} = 0.242... < 1/2, so a(2) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Henk Bruin and Robbert Fokkink, On the records and zeros of a deterministic random walk, arXiv:2503.11734 [math.DS], 2025. See pp. 1, 4.

Cf. A120749, A120750, A120751.

z = 150; r = Sqrt[2]; f[n_] := If[FractionalPart[n*r] < 1/2, 0, 1]
Flatten[Position[Table[f[n], {n, 1, z}], 0]] (* A120243 *)
Flatten[Position[Table[f[n], {n, 1, z}], 1]] (* A120749 *)

Updated by Clark Kimberling, Sep 16 2014

A120749 Numbers k such that {k* sqrt(2)} > 1/2, where { } = fractional part.

Original entry on oeis.org

2, 4, 7, 9, 11, 12, 14, 16, 19, 21, 23, 24, 26, 28, 31, 33, 36, 38, 40, 41, 43, 45, 48, 50, 52, 53, 55, 57, 60, 62, 64, 65, 67, 69, 70, 72, 74, 77, 79, 81, 82, 84, 86, 89, 91, 93, 94, 96, 98, 101, 103, 106, 108, 110, 111, 113, 115, 118, 120, 122, 123, 125, 127, 130, 132, 134

Offset: 1

Clark Kimberling, Jul 01 2006

nonn

The complement of A120749 is A120243.

			Call the present sequence b and its complement a. Then
{r} = {1.4142...} = 0.4142... < 1/2, so a(1) = 1;
{2r} = 0.828... > 1/2, so b(1) = 2;
{3r} = 0.242... < 1/2, so a(2) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Henk Bruin and Robbert Fokkink, On the records and zeros of a deterministic random walk, arXiv:2503.11734 [math.DS], 2025. See pp. 1, 3-4.

Cf. A120243, A120750, A120751.

z = 150; r = Sqrt[2]; f[n_] := If[FractionalPart[n*r] < 1/2, 0, 1]
Flatten[Position[Table[f[n], {n, 1, z}], 0]] (* A120243 *)
Flatten[Position[Table[f[n], {n, 1, z}], 1]] (* A120749 *)
Select[Range[200],FractionalPart[# Sqrt[2]]>1/2&] (* Harvey P. Dale, Aug 20 2024 *)

Updated by Clark Kimberling, Sep 16 2014

A120751 (v(1), u(1), v(2), u(2), v(3), u(3), ...), where u = A120243 and v = A120749.

Original entry on oeis.org

2, 1, 4, 3, 7, 5, 9, 6, 11, 8, 12, 10, 14, 13, 16, 15, 19, 17, 21, 18, 23, 20, 24, 22, 26, 25, 28, 27, 31, 29, 33, 30, 36, 32, 38, 34, 40, 35, 41, 37, 43, 39, 45, 42, 48, 44, 50, 46, 52, 47, 53, 49, 55, 51, 57, 54, 60, 56, 62, 58, 64, 59, 65, 61, 67, 63, 69, 66, 70, 68, 72, 71

Offset: 1

Clark Kimberling, Jul 01 2006

nonn

A permutation of the positive integers.

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

Cf. A120243, A120749, A120750.

z = 4100; r = Sqrt[2];
f[n_] := f[n] = If[FractionalPart[n*r] < 1/2, 0, 1]
u = Flatten[Position[Table[f[n], {n, 1, z}], 0]]; (* A120243 *)
v = Flatten[Position[Table[f[n], {n, 1, z}], 1]]; (* A120749 *)
w = Riffle[Take[v, z/4], Take[u, z/4]]; (* A120751 *)

Updated by Clark Kimberling, Sep 16 2014

cp's OEIS Frontend

A120243 Numbers k such that {k*sqrt(2)} < 1/2, where { } = fractional part.

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A120749 Numbers k such that {k* sqrt(2)} > 1/2, where { } = fractional part.

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A120751 (v(1), u(1), v(2), u(2), v(3), u(3), ...), where u = A120243 and v = A120749.

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