cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A194405 Numbers m such that Sum_{k=1..m} (<1/2 + k*e> - ) < 0 where < > denotes fractional part.

Original entry on oeis.org

1, 5, 33, 37, 65, 69, 71, 72, 73, 75, 76, 77, 79, 83, 97, 101, 103, 104, 105, 107, 108, 109, 111, 115, 129, 133, 135, 136, 137, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 161, 165, 167, 168, 169, 171, 172, 173
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = E; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]       (* A194405 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]       (* A194406 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t3, 1]]       (* A194407 *)

A194406 Numbers m such that Sum_{k=1..m} (<1/2 + k*e> - ) = 0 where < > denotes fractional part.

Original entry on oeis.org

2, 4, 6, 8, 12, 26, 30, 32, 34, 36, 38, 40, 44, 58, 62, 64, 66, 68, 70, 74, 78, 80, 82, 84, 86, 90, 94, 96, 98, 100, 102, 106, 110, 112, 114, 116, 118, 122, 126, 128, 130, 132, 134, 138, 152, 156, 158, 160, 162, 164, 166, 170, 184, 188, 190, 192, 194, 196
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Every term is even; see A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = E; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]       (* A194405 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]       (* A194406 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t3, 1]]       (* A194407 *)

A194407 Numbers m such that Sum_{k=1..m} (<1/2 + k*e> - ) > 0 where < > denotes fractional part.

Original entry on oeis.org

3, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 31, 35, 39, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 63, 67, 81, 85, 87, 88, 89, 91, 92, 93, 95, 99, 113, 117, 119, 120, 121, 123, 124, 125, 127
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = E; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]       (* A194405 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]       (* A194406 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t3, 1]]       (* A194407 *)

A194408 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=Pi and < > denotes fractional part.

Original entry on oeis.org

7, 13, 14, 15, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Pi; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]         (* A194408 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];
Flatten[Position[t2, 1]]         (* A194409 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];
Flatten[Position[t3, 1]]         (* A194410 *)

A194409 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=Pi and < > denotes fractional part.

Original entry on oeis.org

6, 8, 12, 16, 18, 24, 88, 94, 96, 100, 104, 106, 112, 114, 118, 122, 124, 130, 208, 214, 216, 220, 224, 228, 230, 236, 328, 334, 336, 342, 448
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Every term is even; see A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Pi; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]         (* A194408 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];
Flatten[Position[t2, 1]]         (* A194409 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];
Flatten[Position[t3, 1]]         (* A194410 *)

A194410 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=Pi and < > denotes fractional part.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 11, 17, 95, 101, 102, 103, 107, 108, 109, 110, 111, 115, 116, 117, 123, 215, 221, 222, 223, 229, 335
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Pi; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]         (* A194408 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];
Flatten[Position[t2, 1]]         (* A194409 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];
Flatten[Position[t3, 1]]         (* A194410 *)

A194419 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) < 0, where r=sqrt(5) and < > denotes fractional part.

Original entry on oeis.org

4, 5, 8, 59, 76, 77, 80, 131, 148, 149, 152, 203, 220, 221, 224, 275, 292, 293, 296, 686, 758, 830, 902, 974, 991, 992, 995
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[5]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 1000}];
Flatten[Position[t1, 1]]         (* A194419 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 700}];
Flatten[Position[t2, 1]]         (* A194420 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]         (* A194421 *)

A194420 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(5) and < > denotes fractional part.

Original entry on oeis.org

3, 6, 9, 12, 21, 42, 60, 63, 72, 75, 78, 81, 84, 93, 114, 132, 135, 144, 147, 150, 153, 156, 165, 186, 204, 207, 216, 219, 222, 225, 228, 237, 258, 276, 279, 288, 291, 294, 297, 300, 309, 381, 453, 525, 597, 669, 687, 690
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Every term is divisible by 3; see A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[5]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 1000}];
Flatten[Position[t1, 1]]         (* A194419 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 700}];
Flatten[Position[t2, 1]]         (* A194420 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]         (* A194421 *)

A194421 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) > 0, where r=sqrt(5) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 7, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 79, 82, 83, 85, 86, 87
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[5]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 1000}];
Flatten[Position[t1, 1]]         (* A194419 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 700}];
Flatten[Position[t2, 1]]         (* A194420 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]         (* A194421 *)

A194370 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(2) and < > denotes fractional part.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[2]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]] (* A194368 *)
%/2 (* A194369 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]] (* A194370 *)
Previous Showing 41-50 of 67 results. Next

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