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 1-7 of 7 results.

A153717 Minimal exponents m such that the fractional part of (Pi-2)^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 20, 23, 24, 523, 2811, 3465, 3776, 4567, 6145, 8507, 9353, 19790, 41136, 62097, 72506, 107346
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of (Pi-2)^m is less than the fractional part of (Pi-2)^k for all k, 1<=k
The next such number must be greater than 200000.
a(18) > 300000. - Robert Price, Mar 26 2019

Examples

			a(3)=23, since fract((Pi-2)^23)=0.0260069.., but fract((Pi-2)^k)>=0.1326... for 1<=k<=22; thus fract((Pi-2)^23)<fract((Pi-2)^k) for 1<=k<23.

Crossrefs

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A137994, A153721, A154130.

Programs

  • Mathematica
    $MaxExtraPrecision = 100000;
p = 1; Select[Range[1, 10000],
 If[FractionalPart[(Pi - 2)^#] < p, p = FractionalPart[(Pi - 2)^#];
True] &] (* Robert Price, Mar 26 2019 *)

Formula

Recursion: a(1):=1, a(k):=min{ m>1 | fract((Pi-2)^m) < fract((Pi-2)^a(k-1))}, where fract(x) = x-floor(x).

A153705 Greatest number m such that the fractional part of e^A153701(n) <= 1/m.

Original entry on oeis.org

1, 2, 11, 11, 23, 28, 69, 85, 115, 964, 1153, 1292, 1296, 1877, 34015, 156075, 952945
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(3)=11 since 1/12 < fract(e^A153701(3)) = fract(e^3) = 0.0855... <= 1/11.

Crossrefs

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A154130, A153713, A153721.
Cf. A000149.

Programs

  • Mathematica
    A153701 = {1, 2, 3, 9, 29, 45, 75, 135, 219, 732, 1351, 3315, 4795,
   4920, 5469, 28414, 37373};
Table[fp = FractionalPart[E^A153701[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153701]}] (* Robert Price, Mar 25 2019 *)

Formula

a(n) = floor(1/fract(e^A153701(n))), where fract(x) = x-floor(x).

A153713 Greatest number m such that the fractional part of Pi^A137994(n) <= 1/m.

Original entry on oeis.org

7, 159, 270, 307, 744, 757, 796, 1079, 1226, 7804, 13876, 62099, 70718, 86902, 154755
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=159 since 1/160<fract(Pi^A137994(2))=fract(Pi^3)=0.0062766...<=1/159.

Crossrefs

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A137994, A154130, A153721.
Cf. A001672.

Programs

  • Mathematica
    A137994 = {1, 3, 81, 264, 281, 472, 1147, 2081, 3207, 3592, 10479, 12128, 65875, 114791, 118885};
Table[fp = FractionalPart[Pi^A137994[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A137994]}] (* Robert Price, Mar 26 2019 *)

Formula

a(n) = floor(1/fract(Pi^A137994(n))), where fract(x) = x-floor(x).

Extensions

a(14)-a(15) from Robert Price, Mar 26 2019

A153697 Greatest number m such that the fractional part of (10/9)^A153693(n) <= 1/m.

Original entry on oeis.org

9, 11, 30, 82, 6131, 26735, 29430, 76172, 151439, 227416, 771341, 2712159, 4490404
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=11 since 1/12 < fract((10/9)^A153693(2)) = fract((10/9)^7) = 0.09075... <= 1/11.

Crossrefs

Cf. A081464, A153669, A153677, A153685, A153693, A154130, A153705, A153713, A153721.

Programs

  • Mathematica
    A153693 = {1, 7, 50, 62, 324, 3566, 66877, 108201, 123956, 132891,
   182098, 566593, 3501843};
Table[fp = FractionalPart[(10/9)^A153693[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153693]}] (* Robert Price, Mar 25 2019 *)

Formula

a(n) = floor(1/fract((10/9)^A153693(n))), where fract(x) = x-floor(x).

Extensions

a(12)-a(13) from Robert Price, Mar 25 2019

A153689 Greatest number m such that the fractional part of (11/10)^A153685(n) <= 1/m.

Original entry on oeis.org

10, 18, 253, 618, 6009, 6767, 21386, 697723, 4186162, 31102351
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=18 since 1/19 < fract((11/10)^A153685(2)) = fract((11/10)^17) = 0.0544... <= 1/18.

Crossrefs

Cf. A081464, A153669, A153677, A153685, A154130, A153697, A153705, A153713, A153721.

Programs

  • Mathematica
    A153685 = {1, 17, 37, 237, 599, 615, 6638, 13885, 1063942, 9479731};
Table[fp = FractionalPart[(11/10)^A153685[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153685]}] (* Robert Price, Mar 25 2019 *)

Formula

a(n) = floor(1/fract((11/10)^A153685(n))), where fract(x) = x-floor(x).

Extensions

a(9)-a(10) from Robert Price, Mar 25 2019

A153673 Greatest number m such that the fractional part of (101/100)^A153669(n) <= 1/m.

Original entry on oeis.org

100, 147, 703, 932, 1172, 3389, 7089, 8767, 11155, 17457, 20810, 25355, 1129226, 1741049, 1960780, 2179637, 2859688, 11014240, 75249086, 132665447, 499298451
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=147 since 1/148<fract((101/100)^A153669(2))=fract((101/100)^70)=0.00676...<=1/147.

Crossrefs

Cf. A154130, A153669, A153665, A153681, A153689, A153697, A153705, A153713, A153721.

Programs

  • Mathematica
    A153669 = {1, 70, 209, 378, 1653, 2697, 4806, 13744, 66919, 67873,
   75666, 81125, 173389, 529938, 1572706, 4751419, 7159431, 7840546,
   15896994, 71074288, 119325567};
Table[fp = FractionalPart[(101/100)^A153669[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153669]}] (* Robert Price, Mar 25 2019 *)

Formula

a(n) = floor(1/fract((101/100)^A153669(n))), where fract(x) = x-floor(x).

Extensions

a(15)-a(21) from Robert Price, Mar 25 2019

A153681 Greatest number m such that the fractional part of (1024/1000)^A153677(n) <= 1/m.

Original entry on oeis.org

41, 60, 76, 116, 233, 463, 718, 1350, 12472, 13733, 17428, 27955, 32276, 41155, 62437, 69643, 111085, 811799, 2656810, 11462221, 56414953
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=60 since 1/61 < fract((1024/1000)^A153677(2)) = fract((1024/1000)^68) = 0.0164... <= 1/60.

Crossrefs

Cf. A081464, A153669, A153677, A154130, A153689, A153697, A153705, A153713, A153721.

Programs

  • Mathematica
    A153677 = {1, 68, 142, 341, 395, 490, 585, 1164, 1707, 26366, 41358,
   46074, 120805, 147332, 184259, 205661, 385710, 522271, 3418770,
   3675376, 9424094};
Table[fp = FractionalPart[(1024/1000)^A153677[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153677]}] (* Robert Price, Mar 25 2019 *)

Formula

a(n) = floor(1/fract((1024/1000)^A153677(n))), where fract(x) = x-floor(x).

Extensions

a(18)-a(21) from Robert Price, Mar 25 2019
Showing 1-7 of 7 results.

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