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.

Previous Showing 11-17 of 17 results.

A131975 Numbers n where |sinc(n)| decreases monotonically to 0 (where sinc(x)=sin(x)/x).

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 12, 13, 16, 19, 22, 44, 66, 88, 110, 132, 154, 176, 179, 201, 223, 245, 267, 289, 311, 333, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 3550, 3905, 4260, 4615, 4970, 5325, 5680, 6035, 6390, 6745, 7100, 7455, 7810, 8165, 8520, 8875
Offset: 1

Views

Author

Laurent A. Guerin (laurent.a.guerin(AT)orange.fr), Oct 06 2007

Keywords

Crossrefs

Cf. A004112, A046947.

Programs

  • Mathematica
    a = {0, 1}; For[n = 2, n < 10000, n++, If[Abs[Sin[n]/n] < Abs[Sin[a[[ -1]]]/a[[ -1]]], AppendTo[a, n]]]; a (* Stefan Steinerberger, Oct 08 2007 *)
  • PARI
    A131975(nmax)={ local(n=1,aprev=1) ; print1(0) ; while(nA131975(16000) ; \\ R. J. Mathar, Oct 07 2007

Extensions

More terms from R. J. Mathar and Stefan Steinerberger, Oct 07 2007

A224365 a(n) = A063674(n+1) - A063674(n).

Original entry on oeis.org

10, 3, 3, 3, 157, 22, 22, 22, 22, 22, 22, 22, 22, 51808, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355
Offset: 1

Views

Author

Paul Curtz, Apr 09 2013

Keywords

Comments

The repeated terms (3, 22, 355, 5419351, ... from A063674) are the numerators of fractions (3/1, 22/7, 355/113, 5419351/1725033, ...) leading to Pi.
Zu Chongzhi (5th century) discovered 22/7 and 355/113. Adriaan Anthonisz Metius rediscovered 355/113 in 1585.
First differences of A063673 give the denominators: 3, 1, 1, 1, 50, 7, 7, 7, 7, 7, 7, 7, 7, 16489, 113, 113, ... .
Hence 10/3, 157/50, 51808/16489, ... .

Links

Crossrefs

Cf. A063673, A063674.
Cf. A046947, A072398, A002485. A132049, A003077, A068028, A068079.

Programs

  • Mathematica
    A224365 = Reap[ For[ delta0 = 1; d = 1, d < 10^5, d++, delta = Abs[Pi - Round[Pi*d]/d]; If[ delta < delta0, Sow[ Round[Pi*d]]; delta0 = delta]]][[2, 1]] // Differences (* Jean-François Alcover, Apr 10 2013 *)

Formula

a(n) = A063674(n+1) - A063674(n).

A358520 Nearest integer to n/sin(n).

Original entry on oeis.org

1, 2, 21, -5, -5, -21, 11, 8, 22, -18, -11, -22, 31, 14, 23, -56, -18, -24, 127, 22, 25, -2486, -27, -27, -189, 34, 28, 103, -44, -30, -77, 58, 33, 64, -82, -36, -57, 128, 40, 54, -258, -46, -52, 2486, 53, 51, 380, -62, -51, -191, 76, 53, 134, -97
Offset: 1

Views

Author

Bence Bernáth, Nov 20 2022

Keywords

Comments

It is also the nearest integer to sinc(x)^(-1) function.

Examples

			For n=3, 3/sin(3) = 21.25..., therefore a(3) = 21.

Links

Crossrefs

Cf. A272695, A051422, A105666.
Cf. A046947 (Values for n where abs(a(n))/n has records).

Programs

  • Mathematica
    Table[Round[n/Sin[n]], {n, 1, 100}]
  • PARI
    a(n) = round(n/sin(n)); \\ Michel Marcus, Nov 20 2022

A046946 Sin(n) decreases monotonically to 0.

Original entry on oeis.org

1, 3, 44, 311, 377, 688, 710, 103638, 104703, 208341, 312689, 833719, 2292816, 3126535, 6565759, 9692294, 10838702, 74724506, 171126416, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 14885392687
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Conjecture: Next terms are 21053343141, 1783366216531, 3587785776203, 148714156295726, 279510437053578. - Giorgos Kalogeropoulos, Feb 21 2023

References

  • Suggested by a question from Alan Walker (Alan_Walker(AT)sabre.com)

Crossrefs

Cf. A046947.

Programs

  • Mathematica
    z={}; current=1; Timing[ Do[ If[ If[ (t=Sin[ n ])>0, t, 1 ]Wouter Meeussen *)

Extensions

More terms from Wouter Meeussen
Further terms from Michel ten Voorde
a(21)-a(24) from Giorgos Kalogeropoulos, Feb 21 2023

A079038 Fractional part of sinh(n) decreases monotonically to zero.

Original entry on oeis.org

1, 3, 45, 75, 135, 259, 863, 1786, 2483, 2538, 5731, 16095, 36692, 94720
Offset: 0

Views

Author

Benoit Cloitre, Feb 01 2003

Keywords

Crossrefs

Cf. A046947 (for abs(sin(n))).

Programs

  • Mathematica
    $MaxExtraPrecision = Infinity; t = 1; Do[s = FractionalPart[Sinh[n]]; If[s < t, Print[n]; t = s;], {n, 1, 10000}]; (* Vaclav Kotesovec, Apr 06 2020 *)
  • PARI
    lista(nn) = {my(b=2, r=1); print1(1); for(n=1, nn, until(frac(sinh(b))Jinyuan Wang, Apr 04 2020

Extensions

a(8)-a(12) from Jinyuan Wang, Apr 03 2020
a(13) from Vaclav Kotesovec, Apr 06 2020

A079039 Fractional part of cosh(n) decreases monotonically to zero.

Original entry on oeis.org

1, 3, 22, 29, 45, 75, 135, 259, 863, 1786, 2483, 2538, 5731, 16095, 36692, 94720
Offset: 0

Views

Author

Benoit Cloitre, Feb 01 2003

Keywords

Crossrefs

Cf. A046947 ( for abs(sin(x)) ).

Programs

  • Mathematica
    $MaxExtraPrecision = 2^16; k = 1; lst = {}; mn = Infinity; While[k < 100001, a = FractionalPart@ Cosh@k; If[a < mn, mn = a; AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Jul 30 2010 *)
  • PARI
    x=1; y=1; a(n)=if(n<0,0,b=y+1; while(frac(cosh(b))>frac(cosh(x)),b++); x=b; y=b; b)

Extensions

Terms 1786, 2483, 2538 from Zak Seidov, Jul 28 2010
a(9) - a(15) from Robert G. Wilson v, Jul 30 2010

A079040 Fractional part of 1/(1-tanh(n)) decreases monotonically to zero.

Original entry on oeis.org

1, 9, 23, 51, 79, 253, 270, 320, 366, 2460, 3844, 14207, 46819, 68493
Offset: 0

Views

Author

Benoit Cloitre, Feb 01 2003

Keywords

Crossrefs

Cf. A046947 (for abs(sin(n))).

Programs

  • Mathematica
    $MaxExtraPrecision = Infinity; t = 1; Do[s = FractionalPart[1/(1 - Tanh[n])]; If[s < t, Print[n]; t = s;], {n, 1, 5000}]; (* Vaclav Kotesovec, Apr 05 2020 *)
  • PARI
    lista(nn) = {my(b=8, r=1); print1(1); for(n=1, nn, until(frac(1/(1-tanh(b)))Jinyuan Wang, Apr 04 2020

Extensions

a(8)-a(10) from Jinyuan Wang, Apr 03 2020
a(11)-a(13) from Vaclav Kotesovec, Apr 05 2020
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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