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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A194754 Number of integers k in 1..n such that {k*Pi} < {n*Pi}, where { } = fractional part.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 3, 5, 7, 9, 11, 13, 1, 4, 7, 10, 13, 16, 19, 1, 5, 9, 13, 17, 21, 25, 1, 6, 11, 16, 21, 26, 31, 1, 7, 13, 19, 25, 31, 37, 1, 8, 15, 22, 29, 36, 43, 1, 9, 17, 25, 33, 41, 49, 1, 10, 19, 28, 37, 46, 55, 1, 11, 21, 31, 41, 51, 61, 1, 12, 23, 34, 45, 56
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194755, A194738.

Programs

  • Mathematica
    r = Pi; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194754 *)
Table[t[n], {n, 1, 100}]   (* A194755 *)

Extensions

Name clarified by Jon E. Schoenfield, Apr 10 2021

A194755 Number of integers k in 1..n such that {k*Pi} > {n*Pi}, where { } = fractional part.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 7, 6, 5, 4, 3, 2, 1, 14, 12, 10, 8, 6, 4, 2, 21, 18, 15, 12, 9, 6, 3, 28, 24, 20, 16, 12, 8, 4, 35, 30, 25, 20, 15, 10, 5, 42, 36, 30, 24, 18, 12, 6, 49, 42, 35, 28, 21, 14, 7, 56, 48, 40, 32, 24, 16, 8, 63, 54, 45, 36, 27, 18, 9, 70, 60, 50, 40, 30, 20
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194754, A194738.

Programs

  • Mathematica
    r = Pi; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194754 *)
Table[t[n], {n, 1, 100}]   (* A194755 *)

Extensions

Name clarified by Jon E. Schoenfield, Apr 10 2021

A194756 Number of k such that {-k*Pi} < {-n*Pi}, where { } = fractional part.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 7, 6, 5, 4, 3, 2, 15, 13, 11, 9, 7, 5, 3, 22, 19, 16, 13, 10, 7, 4, 29, 25, 21, 17, 13, 9, 5, 36, 31, 26, 21, 16, 11, 6, 43, 37, 31, 25, 19, 13, 7, 50, 43, 36, 29, 22, 15, 8, 57, 49, 41, 33, 25, 17, 9, 64, 55, 46, 37, 28, 19, 10, 71, 61, 51, 41, 31
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194757, A194738.

Programs

  • Mathematica
    r = -Pi; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194756 *)
Table[t[n], {n, 1, 100}]   (* A194757 *)

A194758 Number of k such that {k*log(2)} < {n*log(2)}, where { } = fractional part.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 7, 5, 3, 10, 7, 4, 1, 11, 7, 3, 15, 10, 5, 19, 13, 7, 23, 16, 9, 2, 21, 13, 5, 26, 17, 8, 31, 21, 11, 36, 25, 14, 3, 31, 19, 7, 37, 24, 11, 43, 29, 15, 49, 34, 19, 4, 41, 25, 9, 48, 31, 14, 55, 37, 19, 62, 43, 24, 5, 51, 31, 11, 59, 38, 17, 67, 45, 23, 75
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194759, A194738.

Programs

  • Mathematica
    r = Log[2]; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194758 *)
Table[t[n], {n, 1, 100}]   (* A194759 *)

A194759 Number of k such that {k*log(2)} > {n*log(2)}, where { } = fractional part.

Original entry on oeis.org

0, 1, 2, 0, 2, 4, 0, 3, 6, 0, 4, 8, 12, 3, 8, 13, 2, 8, 14, 1, 8, 15, 0, 8, 16, 24, 6, 15, 24, 4, 14, 24, 2, 13, 24, 0, 12, 24, 36, 9, 22, 35, 6, 20, 34, 3, 18, 33, 0, 16, 32, 48, 12, 29, 46, 8, 26, 44, 4, 23, 42, 0, 20, 40, 60, 15, 36, 57, 10, 32, 54, 5, 28, 51, 0, 24, 48
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194758, A194738.

Programs

  • Mathematica
    r = Log[2]; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194758 *)
Table[t[n], {n, 1, 100}]   (* A194759 *)

A194761 Number of k such that {-k*log(2)} > {-n*log(2)}, where { } = fractional part.

Original entry on oeis.org

0, 0, 0, 3, 2, 1, 6, 4, 2, 9, 6, 3, 0, 10, 6, 2, 14, 9, 4, 18, 12, 6, 22, 15, 8, 1, 20, 12, 4, 25, 16, 7, 30, 20, 10, 35, 24, 13, 2, 30, 18, 6, 36, 23, 10, 42, 28, 14, 48, 33, 18, 3, 40, 24, 8, 47, 30, 13, 54, 36, 18, 61, 42, 23, 4, 50, 30, 10, 58, 37, 16, 66, 44, 22, 74
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194760, A194738.

Programs

  • Mathematica
    r = -Log[2]; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194760 *)
Table[t[n], {n, 1, 100}]   (* A194761 *)

A194762 Number of k such that {k*2^(1/3)} < {n*2^(1/3)}, where { } = fractional part.

Original entry on oeis.org

1, 2, 3, 1, 3, 5, 7, 2, 5, 8, 11, 3, 7, 11, 15, 4, 9, 14, 19, 5, 11, 17, 23, 6, 13, 20, 1, 9, 17, 25, 3, 12, 21, 30, 5, 15, 25, 35, 7, 18, 29, 40, 9, 21, 33, 45, 11, 24, 37, 50, 13, 27, 41, 2, 17, 32, 47, 5, 21, 37, 53, 8, 25, 42, 59, 11, 29, 47, 65, 14, 33, 52, 71, 17, 37
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194763, A194738.

Programs

  • Mathematica
    r = 2^(1/3); p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A194762 *)
Table[t[n], {n, 1, 100}]  (* A194763 *)

A194763 Number of k < n such that {k*2^(1/3)} > {n*2^(1/3)}, where { } = fractional part.

Original entry on oeis.org

0, 0, 0, 3, 2, 1, 0, 6, 4, 2, 0, 9, 6, 3, 0, 12, 8, 4, 0, 15, 10, 5, 0, 18, 12, 6, 26, 19, 12, 5, 28, 20, 12, 4, 30, 21, 12, 3, 32, 22, 12, 2, 34, 23, 12, 1, 36, 24, 12, 0, 38, 25, 12, 52, 38, 24, 10, 53, 38, 23, 8, 54, 38, 22, 6, 55, 38, 21, 4, 56, 38, 20, 2, 57, 38, 19, 76
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Links

Crossrefs

Cf. A194762, A194738.

Programs

  • Maple
    N:= 100: # for a(1) .. a(N)
S:= [seq(frac(k*2^(1/3)),k=1..N)]:
compare:= proc(x,y) local z,a,b;
  z:= y - x;
  a:= coeff(z,2^(1/3));
  b:= z - a*2^(1/3);
  2*a^3 + b^3 > 0
end proc:
seq(nops(select(t -> compare(S[n],t),S[1..n-1])), n=1..N); # Robert Israel, Jan 31 2025
  • Mathematica
    r = 2^(1/3); p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A194762 *)
Table[t[n], {n, 1, 100}]  (* A194763 *)

A194764 Number of k such that {-k*2^(1/3)} < {-n*2^(1/3)}, where { } = fractional part.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 1, 7, 5, 3, 1, 10, 7, 4, 1, 13, 9, 5, 1, 16, 11, 6, 1, 19, 13, 7, 27, 20, 13, 6, 29, 21, 13, 5, 31, 22, 13, 4, 33, 23, 13, 3, 35, 24, 13, 2, 37, 25, 13, 1, 39, 26, 13, 53, 39, 25, 11, 54, 39, 24, 9, 55, 39, 23, 7, 56, 39, 22, 5, 57, 39, 21, 3, 58, 39, 20
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194765, A194738.

Programs

  • Mathematica
    r = -2^(1/3); p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A194764 *)
Table[t[n], {n, 1, 100}]  (* A194765 *)

A194765 Number of k such that {-k*2^(1/3)} > {-n*2^(1/3)}, where { } = fractional part.

Original entry on oeis.org

0, 1, 2, 0, 2, 4, 6, 1, 4, 7, 10, 2, 6, 10, 14, 3, 8, 13, 18, 4, 10, 16, 22, 5, 12, 19, 0, 8, 16, 24, 2, 11, 20, 29, 4, 14, 24, 34, 6, 17, 28, 39, 8, 20, 32, 44, 10, 23, 36, 49, 12, 26, 40, 1, 16, 31, 46, 4, 20, 36, 52, 7, 24, 41, 58, 10, 28, 46, 64, 13, 32, 51, 70, 16, 36
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Crossrefs

Cf. A194764, A194738.

Programs

  • Mathematica
    r = -2^(1/3); p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A194764 *)
Table[t[n], {n, 1, 100}]  (* A194765 *)
Previous Showing 21-30 of 33 results. Next

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