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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A194374 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(5) and < > denotes fractional part.

Original entry on oeis.org

4, 8, 12, 16, 72, 76, 80, 84, 88, 144, 148, 152, 156, 160, 216, 220, 224, 228, 232, 288, 292, 296, 300, 304, 1292, 1296, 1300, 1304, 1308, 1364, 1368, 1372, 1376, 1380, 1436, 1440, 1444, 1448, 1452, 1508, 1512, 1516, 1520, 1524, 1580, 1584, 1588, 1592, 1596, 2584, 2588, 2592, 2596
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368, A194375.

Programs

  • Mathematica
    r = Sqrt[5]; 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]]   (* A194374 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]   (* A194375 *)
  • PARI
    isok(m) = my(r=sqrt(5)); sum(k=1, m, frac(1/2+k*r)-frac(k*r)) == 0; \\ Michel Marcus, Jan 31 2023

Extensions

More terms from Michel Marcus, Jan 31 2023

A194375 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(5) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 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, 73, 74
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368, A194375.

Programs

  • Mathematica
    r = Sqrt[5]; 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, 500}];
Flatten[Position[t1, 1]]   (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]   (* A194374 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]   (* A194375 *)

A194376 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(6) and < > denotes fractional part.

Original entry on oeis.org

2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 198, 200, 202, 204, 206, 218, 220, 222, 224, 226, 238, 240, 242, 244, 246, 258, 260, 262, 264, 266, 278, 280, 282, 284, 286, 396, 398, 400
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368, A194377.

Programs

  • Mathematica
    r = Sqrt[6]; 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, 500}];
Flatten[Position[t1, 1]]   (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]   (* A194376 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]   (* A194377 *)

A194377 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(6) and < > denotes fractional part.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368. Although a(n)=A007957(n) for n = 1..70, the number 208, for example, is here but not A007957.

Crossrefs

Cf. A007957, A194368, A194376.

Programs

  • Mathematica
    r = Sqrt[6]; 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, 500}];
Flatten[Position[t1, 1]]   (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]   (* A194376 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]   (* A194377 *)
  • PARI
    is(n)=my(r=sqrt(6),f=x->x-x\1);sum(k=1,n,f(1/2+k*r)-f(k*r))>0 \\ Charles R Greathouse IV, Jul 25 2012

A194385 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(10) and < > denotes fractional part.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 228, 234, 240, 246, 252, 258, 264, 456, 462, 468, 474, 480, 486, 492, 684, 690, 696, 702, 708, 714, 720
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A010467, A194368, A194386.

Programs

  • Mathematica
    r = Sqrt[10]; 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, 300}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]]     (* A194385 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]     (* A194386 *)

A194390 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(12) and < > denotes fractional part.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 28, 30, 32, 34, 36, 38, 40, 56, 58, 60, 62, 64, 66, 68, 84, 86, 88, 90, 92, 94, 96, 112, 114, 116, 118, 120, 122, 124, 140, 142, 144, 146, 148, 150, 152, 168, 170, 172, 174, 176, 178, 180
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

Every term is even; see A194368.

Crossrefs

Cf. A010469, A194368, A194391.

Programs

  • Mathematica
    r = Sqrt[12]; 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, 400}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194390 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194391 *)

A194391 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(12) and < > denotes fractional part.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 33, 35, 37, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 89, 91, 93
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A010469, A194368, A194390.

Programs

  • Mathematica
    r = Sqrt[12]; 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, 400}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194390 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194391 *)

A194469 Values of m for which sqrt(m) is curbed by 1/2; see Comments for "curbed by".

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 16, 17, 18, 20, 25, 26, 30, 36, 37, 38, 39, 41, 42, 49, 50, 52, 54, 55, 56, 64, 65, 66, 68, 70, 72, 81, 82, 84
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Suppose that r and c are real numbers, that 0- : 1<=k<=n}, where < > denotes fractional part. The inequalities s(n)<0, s(n)=0, s(n)>0 yield up to three sequences that partition the set of positive integers, as in the examples cited at A194368. If s(n)>=0 for every n>=1, we say that r is curbed by c. For r=sqrt(m), clearly r is curbed by 1/2 if m is a square. Conjecture: there are infinitely many nonsquare m for which sqrt(m) is curbed by 1/2, and there are infinitely many m for which sqrt(m) is not curbed by 1/2 (see A194470).
The terms shown here for A194469 are conjectured, based on examinations of s(n) for 1<=n<=B for various B>100.

Crossrefs

Cf. A194368.

Programs

A194470 Complement of A194469 (conjectured): numbers m such that sqrt(m) is not curbed by 1/2.

Original entry on oeis.org

3, 7, 8, 11, 13, 14, 15, 19, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 34, 34, 40, 43, 44, 45, 46, 47, 48, 51, 53, 57, 58, 59, 60, 61, 62, 63, 67, 69, 71, 73, 74, 75, 76, 77, 78, 79, 80, 83, 85, 86, 87, 88
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194469.

Crossrefs

Cf. A194469, A194368.

A194461 Numbers m such that Sum_{k=1..m} ( - ) < 0, where r=(1+sqrt(5))/2 and c=(-1+sqrt(5))/4, and < > denotes fractional part.

Original entry on oeis.org

3, 6, 8, 9, 11, 12, 16, 19, 21, 22, 24, 25, 29, 32, 42, 45, 55, 58, 61, 63, 64, 66, 67, 71, 74, 76, 77, 79, 80, 84, 87, 97, 100, 110, 113, 116, 118, 119, 121, 122, 126, 129, 131, 132, 134, 135, 139, 142, 144, 145, 147, 148, 150, 151, 152, 153, 154, 155, 156
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = GoldenRatio; c = (1/2) FractionalPart[r];
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]]  (* A184461 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t3, 1]]  (* A184462 *)
Previous Showing 51-60 of 67 results. Next

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