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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A153712 Numbers k such that the fractional part of Pi^k is greater than 1-(1/k).

Original entry on oeis.org

1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322, 5269, 151710
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract(Pi^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(15) > 300000. - Robert Price, Mar 25 2019

Examples

			a(3) = 15, since fract(Pi^15) = 0.969... > 0.933... = 1 - (1/15), but fract(Pi^k) <= 1 - (1/k) for 3 <= k <= 14.

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153716, A154130, A153720.
Cf. A001672.

Programs

  • Mathematica
    Select[Range[1000], N[FractionalPart[Pi^#], 100] > 1 - (1/#) &]  (* G. C. Greubel, Aug 25 2016 *)

Extensions

a(14) from Robert Price, Mar 25 2019

A153667 Greatest number m such that the fractional part of (3/2)^A153663(n) >= 1-(1/m).

Original entry on oeis.org

2, 2, 2, 2, 3, 14, 31, 33, 69, 137, 222, 318, 901, 1772, 2747, 12347, 16540, 18198, 135794, 222246, 570361, 2134829, 6901329, 75503109, 814558605
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Examples

			a(5)=3, since 1-(1/4)=0.75>fract((3/2)^A153663(5))=fract((3/2)^12)=0.746...>=1-(1/3).

Crossrefs

Cf. A002379, A081464, A153662, A153663, A153664, A153665, A153666, A153668.

Programs

  • Mathematica
    A153663 = {1, 5, 8, 10, 12, 14, 46, 58, 105, 157, 163, 455, 1060, 1256, 2677, 8093, 28277, 33327, 49304, 158643, 164000, 835999, 2242294, 25380333, 92600006};
Table[fp = FractionalPart[(3/2)^A153663[[n]]]; m = Floor[1/(1-fp)];
While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153663]}] (* Robert Price, Mar 26 2019 *)

Formula

a(n) = floor(1/(1-fract((3/2)^A153663(n)))), where fract(x) = x-floor(x).

Extensions

a(22)-a(25) from Robert Price, May 10 2012

A153708 Greatest number m such that the fractional part of e^A153704(n) >= 1-(1/m).

Original entry on oeis.org

3, 23, 27, 261, 348, 2720, 72944, 347065, 244543
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2) = 23, since 1-(1/24) = 0.9583... > fract(e^A153704(2)) = fract(e^8) = 0.95798... >= 0.95652... >= 1-(1/23).

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A154130, A153716, A153724.
Cf. A000149.

Programs

  • Mathematica
    A153704 = {1, 8, 19, 178, 209, 1907, 32653, 119136, 220010};
Table[fp = FractionalPart[E^A153704[[n]]]; m = Floor[1/fp];
While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153704]}] (* Robert Price, May 10 2019 *)

Formula

a(n) = floor(1/(1-fract(e^A153704(n)))), where fract(x) = x-floor(x).

Extensions

a(8)-a(9) from Robert Price, May 10 2019

A153716 Greatest number m such that the fractional part of Pi^A153712(n) >= 1-(1/m).

Original entry on oeis.org

1, 7, 32, 53, 189, 131, 2665, 10810, 2693, 1976, 3697, 4289, 26577, 483367
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153712(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153712, A154130, A153724.
Cf. A001672.

Programs

  • Mathematica
    A153712 = {1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322,
   5269, 151710};
Table[Floor[1/(1 - FractionalPart[Pi^A153712[[n]]])], {n, 1,
Length[A153712]}] (* Robert Price, May 10 2019 *)

Formula

a(n) = floor(1/(1-fract(Pi^A153712(n)))), where fract(x) = x-floor(x).

Extensions

a(14) from Robert Price, May 10 2019

A153720 Numbers k such that the fractional part of (Pi-2)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 5, 8, 85, 911, 2921, 4491, 11543, 15724, 27683, 29921, 37276, 126659
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract((Pi-2)^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 200000.
a(14) > 300000. - Robert Price, Mar 26 2019

Examples

			a(3) = 8, since fract((Pi-2)^8) = 0.8846247315... > 0.875 = 1 - (1/8), but fract((Pi-2)^k) = 0.2134..., 0.5268... <= 1 - (1/k) for 6 <= k <= 7.

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153712, A153724, A154130.

Programs

  • Mathematica
    Select[Range[1000], N[FractionalPart[(Pi - 2)^#], 100] > 1 - (1/#) &] (* G. C. Greubel, Aug 25 2016 *)

A153724 Greatest number m such that the fractional part of (Pi-2)^A153720(n) >= 1-(1/m).

Original entry on oeis.org

1, 16, 8, 158, 946, 8786, 16159, 20188, 61392, 34039, 31425, 59154, 217556
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(4)=158, since 1-(1/159) = 0.993710... > fract((Pi-2)^A153720(4)) = fract(Pi^85) = 0.993693... >= 0.993670... = 1-(1/158).

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153712, A153720, A154130.

Programs

  • Mathematica
    A153720 = {1, 5, 8, 85, 911, 2921, 4491, 11543, 15724, 27683, 29921,
   37276, 126659};
Table[Floor[1/(1 - FractionalPart[(Pi - 2)^A153720[[n]]])], {n, 1,
Length[A153720]}] (* Robert Price, May 10 2019 *)

Formula

a(n):=floor(1/(1-fract((Pi-2)^A153720(n)))), where fract(x) = x-floor(x).

A153684 Greatest number m such that the fractional part of (1024/1000)^A153680(n) >= 1-(1/m).

Original entry on oeis.org

1, 93, 123, 1061, 395, 1360, 4137, 2706, 66910, 21740, 15986, 58999, 571666, 1192010, 793642, 1093343, 3476524
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=93, since 1-(1/94)=0.98936...>fract((1024/1000)^A153680(2))=fract((1024/1000)^29)=0.98929...>=1-(1/93).

Crossrefs

Cf. A153664, A153672, A153680, A154130, A153692, A153700, A153708, A153716, A153724.

Formula

a(n):=floor(1/(1-fract((1024/1000)^A153680(n)))), where fract(x) = x-floor(x).

Extensions

a(16) - a(17) from Hagen von Eitzen, May 16 2009

A153700 Greatest number m such that the fractional part of (10/9)^A153696(n) >= 1-(1/m).

Original entry on oeis.org

1, 8, 15, 264, 8741, 15912, 409895
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(3)=15, since 1-(1/16)=0.9375>fract((10/9)^A153696(3))=fract((10/9)^13)=0.9341...>=1-(1/15).

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A154130, A153708, A153716, A153724.

Formula

a(n):=floor(1/(1-fract((10/9)^A153696(n)))), where fract(x) = x-floor(x).

A153692 Greatest number m such that the fractional part of (11/10)^A153688(n) >= 1-(1/m).

Original entry on oeis.org

1, 19, 151, 200, 709, 5727, 15908, 162819, 120479, 109526, 302991
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=19, since 1-(1/20)=0.95>fract((11/10)^A153688(2))=fract((11/10)^7)=0.9487...>=1-(1/19).

Crossrefs

Cf. A153664, A153672, A153680, A153684, A154130, A153700, A153708, A153716, A153724.

Formula

a(n):=floor(1/(1-fract((11/10)^A153688(n)))), where fract(x) = x-floor(x).

A153676 Greatest number m such that the fractional part of (101/100)^A153672(n) >= 1-(1/m).

Original entry on oeis.org

1, 76, 238, 913, 1334, 4645, 17396, 351085, 69587, 552184, 329808, 381654, 35874097, 5011174, 6220178, 33773592, 13149134, 105749940
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=76, since 1-(1/77)=0.9870...>fract((101/100)^A153672(2))=fract((101/100)^69)=0.98689...>=1-(1/76).

Crossrefs

Cf. A153664, A154130, A153668, A153684, A153692, A153700, A153708, A153716, A153724.

Formula

a(n):=floor(1/(1-fract((101/100)^A153672(n)))), where fract(x) = x-floor(x).

Extensions

a(13)-a(18) from Robert Price, May 10 2012
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