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.

Showing 1-10 of 10 results.

A086868 a(n) = Floor(1/sin(A070752(n))).

Original entry on oeis.org

1, 1, 7, 1, 1, 2, 2, 1, 1, 6, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 56, 1, 1, 8, 1, 1, 2, 2, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 28, 1, 1, 9, 1, 1, 2, 2, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 18, 1, 1, 11, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 14, 1, 1, 14, 1, 1, 2, 2
Offset: 1

Views

Author

Philippe Deléham, Aug 20 2003

Keywords

A070751 Numbers n such that sin(n) < 0.

Original entry on oeis.org

4, 5, 6, 10, 11, 12, 16, 17, 18, 22, 23, 24, 25, 29, 30, 31, 35, 36, 37, 41, 42, 43, 48, 49, 50, 54, 55, 56, 60, 61, 62, 66, 67, 68, 69, 73, 74, 75, 79, 80, 81, 85, 86, 87, 92, 93, 94, 98, 99, 100, 104, 105, 106, 110, 111, 112, 113, 117
Offset: 1

Views

Author

Reinhard Zumkeller, May 04 2002

Keywords

Comments

A070747(a(n)) = -1.

Crossrefs

Cf. A070752, A070753.

Programs

  • Mathematica
    Select[Range[150],Sin[#]<0&] (* Harvey P. Dale, Dec 24 2023 *)

A332480 Numbers k such that sin(k) > 0 and cos(k) > 0.

Original entry on oeis.org

1, 7, 13, 14, 19, 20, 26, 32, 38, 39, 44, 45, 51, 57, 58, 63, 64, 70, 76, 82, 83, 88, 89, 95, 101, 102, 107, 108, 114, 120, 126, 127, 132, 133, 139, 145, 146, 151, 152, 158, 164, 170, 171, 176, 177, 183, 189, 190, 195, 196, 202, 208, 214, 215, 220, 221, 227
Offset: 1

Views

Author

Clark Kimberling, Apr 14 2020

Keywords

Comments

The sequences A332480-A332483 partition the positive integers.
Subsequence of A327422, where sin(k)*cos(k)>0. - R. J. Mathar, Apr 27 2020

Crossrefs

Cf. A070752, A131503, A246442, A246443, A246450, A246451, A332481-A332487.

Programs

  • Mathematica
    Select[Range[300], Sin[#] > 0 && Cos[#] > 0 &] (* A332480 *)
Select[Range[300], Sin[#] > 0 && Cos[#] < 0 &] (* A332481 *)
Select[Range[300], Sin[#] < 0 && Cos[#] > 0 &] (* A332482 *)
Select[Range[300], Sin[#] < 0 && Cos[#] < 0 &] (* A332483 *)

A070747 a(n) = signum(sin(n)), where signum=A057427.

Original entry on oeis.org

0, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1
Offset: 0

Views

Author

Reinhard Zumkeller, May 04 2002

Keywords

Comments

a(n) <> 0 for n>0.

Examples

			For n=10: sin(10) = sin(10-2*Pi) < 0, as Pi < 10-2*Pi < 2*Pi, therefore a(10) = signum(sin(10)) = -1.

Crossrefs

Cf. A000493, A000494, A032615, A070748, A070751, A070752.

Programs

Formula

a(n) = (-1)^A032615(n) for n>0. - Michel Marcus, Mar 20 2021

A327422 Positive integers k such that tan(k) > 0 (or equivalently, cot(k) > 0).

Original entry on oeis.org

1, 4, 7, 10, 13, 14, 16, 17, 19, 20, 22, 23, 26, 29, 32, 35, 36, 38, 39, 41, 42, 44, 45, 48, 51, 54, 57, 58, 60, 61, 63, 64, 66, 67, 70, 73, 76, 79, 80, 82, 83, 85, 86, 88, 89, 92, 95, 98, 101, 102, 104, 105, 107, 108, 110, 111, 114, 117, 120, 123, 124, 126, 127, 129
Offset: 1

Views

Author

Jianing Song, Nov 29 2019

Keywords

Comments

Complement of A327423.
Numbers k such that 0 < {k/Pi} < 1/2, where {} is the fractional part.
Numbers k such that A330035(k) = +1.
Also positive integers k such that sin(2k) > 0, i.e., k such that 2k is in A070752.

Links

Crossrefs

Cf. A327423, A330035.
Cf. A070752 (sin(k) > 0 or csc(k) > 0), A070751 (sin(k) < 0 or csc(k) < 0), A131503 (cos(k) > 0 or sec(k) > 0), this sequence (tan(k) > 0 or cot(k) > 0), A327423 (tan(k) < 0 or cot(k) < 0).

Programs

  • Magma
    [n : n in[1..300]|Tan(n) gt 0]; // K. D. Bajpai, Dec 15 2019
  • Mathematica
    upto[n_] := Select[Range[n], Tan[#] > 0 &]
A327422={}; Do[ If[Tan[n] > 0, AppendTo[A327422, n]],{n, 200}]; A327422 (* K. D. Bajpai, Dec 15 2019 *)
Select[Range[130],Tan[#]>0&] (* Harvey P. Dale, Aug 26 2021 *)
  • PARI
    A327422_up_to_n(n) = my(v=vector(n,k,k)); select(k->tan(k)>0,v)

A327423 Positive integers k such that tan(k) < 0 (or equivalently, cot(k) < 0).

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 15, 18, 21, 24, 25, 27, 28, 30, 31, 33, 34, 37, 40, 43, 46, 47, 49, 50, 52, 53, 55, 56, 59, 62, 65, 68, 69, 71, 72, 74, 75, 77, 78, 81, 84, 87, 90, 91, 93, 94, 96, 97, 99, 100, 103, 106, 109, 112, 113, 115, 116, 118, 119, 121, 122, 125, 128
Offset: 1

Views

Author

Jianing Song, Nov 29 2019

Keywords

Comments

Complement of A327422.
Numbers k such that 1/2 < {k/Pi} < 1, where {} is the fractional part.
Numbers k such that A330035(k) = -1.
Also positive integers k such that sin(2k) < 0, i.e., k such that 2k is in A070751.

Crossrefs

Cf. A330035.
Cf. A070752 (sin(k) > 0 or csc(k) > 0), A070751 (sin(k) < 0 or csc(k) < 0), A131503 (cos(k) > 0 or sec(k) > 0), A327422 (tan(k) > 0 or cot(k) > 0), this sequence (tan(k) < 0 or cot(k) < 0).

Programs

  • Mathematica
    upto[n_] := Select[Range[n], Tan[#] < 0 &]
  • PARI
    A327423_up_to_n(n) = my(v=vector(n,k,k)); select(k->tan(k)<0,v)

A070754 Primes p such that sin(p) > 0.

Original entry on oeis.org

2, 3, 7, 13, 19, 47, 53, 59, 71, 83, 89, 97, 101, 103, 107, 109, 127, 139, 151, 179, 191, 197, 223, 227, 229, 233, 239, 241, 271, 277, 283, 311, 317, 347, 353, 359, 367, 373, 379, 397, 409, 421, 449, 461, 467, 479, 491, 499, 503
Offset: 1

Views

Author

Reinhard Zumkeller, May 04 2002

Keywords

Comments

A070748(A049084(a(n))) = A070747(a(n)) = 1.

Crossrefs

Cf. A070753, A070748, A002144 (sin((Pi/2)*p) > 0 instead of sin(p) > 0), A070752.

Programs

A332481 Numbers k such that sin(k) > 0 and cos(k) < 0.

Original entry on oeis.org

2, 3, 8, 9, 15, 21, 27, 28, 33, 34, 40, 46, 47, 52, 53, 59, 65, 71, 72, 77, 78, 84, 90, 91, 96, 97, 103, 109, 115, 116, 121, 122, 128, 134, 135, 140, 141, 147, 153, 159, 160, 165, 166, 172, 178, 179, 184, 185, 191, 197, 203, 204, 209, 210, 216, 222, 223, 228
Offset: 1

Views

Author

Clark Kimberling, Apr 14 2020

Keywords

Comments

The sequences A332480-A332483 partition the positive integers.

Crossrefs

Cf. A070752, A246444, A332480, A332482, A332483. Subsequence of A327423.

Programs

  • Mathematica
    Select[Range[300], Sin[#] > 0 && Cos[#] > 0 &] (* A332480 *)
Select[Range[300], Sin[#] > 0 && Cos[#] < 0 &] (* A332481 *)
Select[Range[300], Sin[#] < 0 && Cos[#] > 0 &] (* A332482 *)
Select[Range[300], Sin[#] < 0 && Cos[#] < 0 &] (* A332483 *)

A342680 Decimal expansion of Sum_{n>=1} sin(sin(n)/n).

Original entry on oeis.org

9, 6, 1, 3, 9, 4, 3, 1, 5, 9, 4, 5, 7, 3, 6, 5, 4, 7, 2, 4, 7, 6, 4, 5, 9, 5, 3, 1, 6, 1, 5, 4, 7, 3, 0, 6, 8, 6, 8, 5, 8, 2, 6, 9, 3, 0, 1, 0, 5, 8, 4, 6, 0, 4, 5, 5, 1, 1, 5, 1, 4, 9, 1, 8, 1, 8, 6, 3, 3, 7, 8, 0, 2, 9, 1, 4, 6, 9, 9, 7, 0, 6, 6, 7, 5, 4, 2, 4, 3, 2, 5, 5, 4, 9, 5, 5, 5, 5, 2, 6, 9, 8, 7, 9, 2
Offset: 0

Views

Author

Bernard Schott, Mar 18 2021

Keywords

Comments

Abel summation shows the series is convergent.

Examples

			0.96139431594573654724764595316154730686858269301058...

References

  • Konrad Knopp, Theory and Application of Infinite Series, Blackie, 1928, p. 313.
  • Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice C.3.7 2.3.b)4. p. 309.

Links

Crossrefs

Cf. A070751, A070752, A096418, A248946.

Programs

  • Magma
    nDgtsOutput:=110; nDgtsPrecision:=nDgtsOutput+10; SetDefaultRealField(RealField(nDgtsPrecision)); kMax:=Ceiling(1.395*nDgtsPrecision-3); mMax:=Ceiling(1.5*kMax); sum:=0.0; S1:=[0.0 : j in [1..kMax]]; n:=0; for m in [1..mMax] do S2:=S1; for k in [1..355] do n:=n+1; sum+:=Sin(Sin(n)/n); end for; S1[1]:=sum; for j in [1..kMax-1] do S1[j+1]:=(S2[j]+S1[j])/2; end for; end for; ChangePrecision(S1[#S1], nDgtsOutput); // The constants 1.395 and 1.5 were empirically derived; 355 is used because 355/Pi is very close to an odd integer. - Jon E. Schoenfield, Mar 21 2021

Extensions

a(3)-a(104) from Jon E. Schoenfield, Mar 20 2021

A332488 a(n) = least positive integer k such that sin(n*k)*sin(n*k + k) < 0.

Original entry on oeis.org

3, 1, 1, 1, 2, 11, 4, 1, 1, 1, 2, 5, 7, 2, 1, 1, 1, 3, 20, 2, 1, 1, 1, 2, 23, 3, 1, 1, 1, 2, 7, 5, 1, 1, 1, 1, 4, 10, 2, 1, 1, 1, 3, 177, 3, 1, 1, 1, 2, 11, 4, 1, 1, 1, 2, 5, 6, 2, 1, 1, 1, 3, 18, 2, 1, 1, 1, 2, 27, 3, 1, 1, 1, 2, 7, 5, 1, 1, 1, 1, 4, 9, 2
Offset: 1

Views

Author

Clark Kimberling, Apr 21 2020

Keywords

Comments

a(n) = least positive integer k such that sin(n*k) and sin(n*k + k) have opposite signs.

Examples

			The signs of sin(6), sin(12), sin(18), ..., sin(72) are indicated by - - - - - - - - - - - + ; that's eleven -'s followed by +, so that a(6) = 11.

Crossrefs

Cf. A070751, A070752, A332489, A332499.

Programs

  • Mathematica
    Table[First[Map[Length, Split[Table[Sign[Sin[k n]], {k, 1, 500}]]]], {n, 1, 100}]
Showing 1-10 of 10 results.

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