A246356 -id:A246356 - cp's OEIS frontend

A247454 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(3)}, and { } = fractional part.

3, 5, 6, 7, 9, 12, 16, 17, 19, 20, 22, 23, 24, 28, 29, 30, 32, 33, 37, 41, 45, 48, 49, 52, 56, 57, 58, 61, 62, 66, 67, 69, 74, 75, 76, 81, 82, 88, 89, 90, 91, 93, 96, 98, 99, 101, 102, 104, 105, 106, 108, 111, 113, 115, 116, 117, 120, 122, 125, 129, 130, 131

Offset: 1

Clark Kimberling, Sep 17 2014

Every positive integer lies in exactly one of the sequences A247454 and A247324. Let s denote either sequence; is lim(#s < n)/n = 1/2, where (#s < n) represents the number of numbers in s that are < n?

			{sqrt(2)} has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,...
{sqrt(3)} has binary digits 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0,..
so that a(1) = 3 and a(2) = 5.

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

Cf. A246356, A247324.

z = 200; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];
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]]  (* A247454 *)
Flatten[Position[t, 0]]  (* A247324 *)

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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 18 2014

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?

			{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.

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

Cf. A246356, A247456, A247457, A247458.

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 *)

A246357 Numbers k such that d(r,k) = 0 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

Original entry on oeis.org

1, 4, 8, 10, 11, 14, 15, 21, 25, 38, 42, 47, 51, 54, 55, 59, 60, 63, 64, 70, 72, 78, 83, 85, 86, 92, 100, 107, 109, 119, 121, 128, 134, 136, 147, 148, 150, 153, 157, 162, 168, 169, 173, 182, 183, 184, 198, 200, 209, 211, 214, 215, 218, 226, 227, 229, 241

Offset: 1

Clark Kimberling, Sep 17 2014

Every positive integer lies in exactly one of these: A246356, A246357, A246358, A247356.

			{sqrt(2)} has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,...
{sqrt(3)} has binary digits 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0,..
so that a(1) = 1 and a(2) = 4.

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

Cf. A247454, A246356, A246358, A247356.

z = 500; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];
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]]  (* A246356 *)
Flatten[Position[t2, 1]]  (* A246357 *)
Flatten[Position[t3, 1]]  (* A246358 *)
Flatten[Position[t4, 1]]  (* A247356 *)

A246358 Numbers k such that d(r,k) = 1 and d(s,k) = 0, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

Original entry on oeis.org

2, 13, 18, 26, 27, 31, 34, 35, 36, 39, 40, 43, 44, 46, 50, 53, 65, 68, 71, 73, 77, 79, 80, 84, 87, 94, 95, 97, 103, 110, 112, 114, 118, 123, 124, 126, 127, 132, 133, 135, 142, 143, 145, 146, 152, 155, 156, 160, 171, 174, 176, 180, 192, 196, 197, 205, 206

Offset: 1

Clark Kimberling, Sep 17 2014

Every positive integer lies in exactly one of these: A246356, A246357, A246358, A247356.

			{sqrt(2)} has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,...
{sqrt(3)} has binary digits 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0,..
so that a(1) = 2 and a(2) = 13.

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

Cf. A247454, A246356, A246357, A247356.

z = 500; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];
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]]  (* A246356 *)
Flatten[Position[t2, 1]]  (* A246357 *)
Flatten[Position[t3, 1]]  (* A246358 *)
Flatten[Position[t4, 1]]  (* A247356 *)

A247356 Numbers k such that d(r,k) = 1 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

Original entry on oeis.org

3, 5, 7, 16, 17, 19, 22, 23, 30, 32, 33, 41, 45, 49, 56, 61, 67, 74, 75, 76, 88, 90, 91, 98, 99, 101, 105, 108, 115, 116, 117, 120, 125, 131, 137, 138, 140, 141, 154, 158, 164, 167, 170, 172, 175, 178, 185, 188, 189, 193, 194, 199, 203, 221, 230, 231, 234

Offset: 1

Clark Kimberling, Sep 17 2014

Every positive integer lies in exactly one of these: A246356, A246357, A246358, A247356.

			{sqrt(2)} has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,...
{sqrt(3)} has binary digits 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0,..
so that a(1) = 3 and a(2) = 5.

Clark Kimberling, Table of n, a(n) for n = 1..1115

Cf. A247454, A246356, A246357, A246358.

z = 500; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];
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]]  (* A246356 *)
Flatten[Position[t2, 1]]  (* A246357 *)
Flatten[Position[t3, 1]]  (* A246358 *)
Flatten[Position[t4, 1]]  (* A247356 *)

A247324 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(3)}, and { } = fractional part.

Original entry on oeis.org

1, 2, 4, 8, 10, 11, 13, 14, 15, 18, 21, 25, 26, 27, 31, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 50, 51, 53, 54, 55, 59, 60, 63, 64, 65, 68, 70, 71, 72, 73, 77, 78, 79, 80, 83, 84, 85, 86, 87, 92, 94, 95, 97, 100, 103, 107, 109, 110, 112, 114, 118, 119

Offset: 1

Clark Kimberling, Sep 17 2014

Every positive integer lies in exactly one of the sequences A247454 and A247324.

			{sqrt(2)} has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1,...
{sqrt(3)} has binary digits 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0,..
so that a(1) = 1 and a(2) = 2.

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

Cf. A246356, A247454.

z = 200; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[3]];
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]]  (* A247454 *)
Flatten[Position[t, 0]]  (* A247324 *)

cp's OEIS Frontend

A247454 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(3)}, and { } = fractional part.

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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.

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Mathematica

A246357 Numbers k such that d(r,k) = 0 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

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Mathematica

A246358 Numbers k such that d(r,k) = 1 and d(s,k) = 0, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

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Mathematica

A247356 Numbers k such that d(r,k) = 1 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(3)}, and { } = fractional part.

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Examples

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Mathematica

A247324 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(3)}, and { } = fractional part.

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Examples

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Mathematica