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 21 results. Next

A153668 Greatest number m such that the fractional part of (3/2)^A153664(n) >= 1-(1/m).

Original entry on oeis.org

2, 14, 222, 1772, 2747, 12347, 135794, 90529, 222246, 570361, 2134829, 6901329, 4600886, 3067257, 5380892, 75503109, 814558605, 543039070, 362026046, 241350697, 160900465, 107266976, 101721580, 190708740, 127139160
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Examples

			a(2)=14, since 1-(1/15)=0.933...>fract((3/2)^A153664(2))=fract((3/2)^14)=0.929...>=1-(1/14).

Crossrefs

Cf. A002379, A081464, A153662, A153663, A153664, A153665, A153666, A153667.

Programs

  • Mathematica
    A153664 = {1, 14, 163, 1256, 2677, 8093, 49304, 49305, 158643, 164000, 835999, 2242294, 2242295, 2242296, 3965133, 25380333, 92600006, 92600007, 92600008, 92600009, 92600010, 92600011, 9267816, 125040717, 125040718};
Table[fp = FractionalPart[(3/2)^A153664[[n]]]; m = Floor[1/(1 - fp)];
While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153664]}] (* Robert Price, Mar 26 2019 *)

Formula

a(n) = floor(1/(1-fract((3/2)^A153664(n)))), where fract(x) = x-floor(x).

Extensions

a(11)-a(25) from Robert Price, May 10 2012

A153662 Numbers k such that the fractional part of (3/2)^k is less than 1/k.

Original entry on oeis.org

1, 2, 4, 7, 3328, 3329, 4097, 12429, 12430, 12431, 18587, 44257, 112896, 129638, 4264691, 144941960, 144941961, 144941962
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Comments

Numbers k such that fract((3/2)^k) < 1/k, where fract(x) = x-floor(x).
The next term is greater than 3*10^8.

Examples

			a(4) = 7 since fract((3/2)^7) = 0.0859375 < 1/7, but fract((3/2)^5)  = 0.59375 >= 1/5 and fract((3/2)^6) = 0.390625 >= 1/6.

Crossrefs

Cf. A002379, A081464, A153663, A153664, A153665, A153666, A153667, A153668.

Programs

  • Mathematica
    Select[Range[1000], FractionalPart[(3/2)^#] < (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

Extensions

a(15)-a(18) from Robert Gerbicz, Nov 21 2010

A153663 Minimal exponents m such that the fractional part of (3/2)^m reaches a maximum (when starting with m=1).

Original entry on oeis.org

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
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Comments

Recursive definition: a(1)=1, a(n) = least number m such that the fractional part of (3/2)^m is greater than the
fractional part of (3/2)^k for all k, 1<=k
The fractional part of k=835999 is .999999 5 which is greater than (k-1)/k. The fractional part of k=2242294 is .999999 8 which is greater than (k-1)/k. The fractional part of k=25380333 is .999999 98 which is greater than (k-1)/k. The fractional part of k=92600006 is .999999 998 which is greater than (k-1)/k. So, all additional numbers in this sequence must be in A153664 and >3*10^8. - Robert Price, May 09 2012

Examples

			a(2)=5, since fract((3/2)^5)=0.59375, but fract((3/2)^k)=0.5, 0.25, 0.375, 0.0625 for 1<=k<=4; thus
fract((3/2)^5)>fract((3/2)^k) for 1<=k<5.

Crossrefs

Cf. A002379, A153661, A153662, A153664, A153665, A153666, A153667, A153668.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = For[m = a[n-1]+1, True, m++, f = FractionalPart[(3/2)^m]; If[AllTrue[Range[m-1], f > FractionalPart[(3/2)^#]&], Print[n, " ", m]; Return[m]]];
Array[a, 21] (* Jean-François Alcover, Feb 25 2019 *)

Formula

Recursion: a(1):=1, a(k):=min{ m>1 | fract((3/2)^m) > fract((3/2)^a(k-1))}, where fract(x) = x-floor(x).

Extensions

a(22)-a(25) from Robert Price, May 09 2012

A153672 Numbers k such that the fractional part of (101/100)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 69, 180, 783, 859, 1803, 10763, 19105, 39568, 50172, 132572, 355146, 1452050, 2245950, 3047334, 3933030, 4165171, 98544173
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract((101/100)^k) > 1-(1/k), where fract(x) = x-floor(x).
The next term is greater than 2*10^8.

Examples

			a(2) = 69, since fract((101/100)^69) = 0.9868... > 0.9855... = 1 - (1/69), but fract((101/100)^k) <= 1 - (1/k) for 1 < k < 69.

Crossrefs

Cf. A153664, A154130, A153676, A153680, A153688, A153696, A153704, A153712, A153720.

Programs

  • Mathematica
    Select[Range[1000], FractionalPart[(101/100)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

Extensions

a(13)-a(18) from Robert Gerbicz, Nov 22 2010

A153680 Numbers k such that the fractional part of (1024/1000)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 29, 82, 134, 277, 1306, 2036, 2349, 6393, 9389, 9816, 21689, 34477, 145984, 171954, 956357, 2746739
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract((1024/1000)^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 5*10^5.

Examples

			a(2) = 29, since fract((1024/1000)^29) = 0.98929... > 0.9655... = 1 - (1/29), but fract((1024/1000)^k) <= 1 - (1/k) for 1 < k < 29.

Crossrefs

Cf. A153664, A153672, A153684, A154130, A153688, A153696, A153704, A153712, A153720.

Programs

  • Mathematica
    Select[Range[1000], FractionalPart[(1024/1000)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

Extensions

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

A153688 Numbers k such that the fractional part of (11/10)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 7, 77, 103, 320, 1821, 2992, 15290, 88651, 88652, 180168, 410498, 548816, 672732
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract((11/10)^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 2*10^5.
a(15) > 10^7. Robert Price, Mar 19 2019

Examples

			a(2) = 7, since fract((11/10)^7) = 0.9487... > 0.8571... = 1 - (1/7), but fract((11/10)^k) <= 1 - (1/k) for 1 < k < 7.

Crossrefs

Cf. A153664, A153672, A153684, A153692, A154130, A153696, A153704, A153712, A153720.

Programs

  • Mathematica
    Select[Range[1000], FractionalPart[(11/10)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

Extensions

a(12)-a(14) from Robert Price, Mar 19 2019

A153696 Numbers k such that the fractional part of (10/9)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 6, 13, 17, 5513, 12746, 126237, 430116, 477190, 1295623, 1319307, 3596185, 6109350
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract((10/9)^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 2*10^5.
a(14) > 10^7. - Robert Price, Mar 24 2019

Examples

			a(3) = 13, since fract((10/9)^13) = 0.9341... > 0.923... = 1 - (1/13), but fract((10/9)^k) <= 1 - (1/k) for 1 < k < 13.

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153700, A154130, A153704, A153712, A153720.

Programs

  • Mathematica
    Select[Range[1000], FractionalPart[(10/9)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

Extensions

a(8)-a(13) from Hans Havermann, Apr 04 2016

A153704 Numbers k such that the fractional part of e^k is greater than 1-(1/k).

Original entry on oeis.org

1, 8, 19, 178, 209, 1907, 32653, 119136, 220010
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Comments

Numbers k such that fract(e^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(10) > 300,000. Robert Price, Mar 23 2019

Examples

			a(2)=8, since fract(e^8) = 0.957987... >0.875 = 1-(1/8), but fract(e^k) = 0.389..., 0.085..., 0.598..., 0.413..., 0.428..., 0.633... for 2<=k<=7 which all are less than 1-(1/k).

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153708, A154130, A153712, A153720.
Cf. A000149.

Programs

  • Mathematica
    Select[Range[2000], N[FractionalPart[E^#], 1000] >= 1 - (1/#) &] (* G. C. Greubel, Aug 25 2016 *)

Extensions

a(8)-a(9) from Robert Price, Mar 23 2019

A153665 Greatest number m such that the fractional part of (3/2)^A081464(n) <= 1/m.

Original entry on oeis.org

2, 4, 16, 25, 89, 91, 105, 127, 290, 668, 869, 16799, 92694, 137921, 257825, 350408, 419427, 723749, 5271294, 14223700, 18090494, 88123482, 706641581
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Examples

			a(4)=25 since 1/26<fract((3/2)^A081464(4))=fract((3/2)^29)=0.039...<=1/25.

Crossrefs

Cf. A002379, A153662, A153663, A153664, A153666, A153667, A153668.

Programs

  • Mathematica
    A081464 = {1, 2, 4, 29, 95, 153, 532, 613, 840, 2033, 2071, 3328, 12429, 112896, 129638, 371162, 1095666, 3890691, 4264691, 31685458, 61365215, 92432200, 144941960};
Table[fp = FractionalPart[(3/2)^A081464[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++];  m - 1, {n, 1, Length[A081464]}] (* Robert Price, Mar 26 2019 *)

Formula

a(n):=floor(1/fract((3/2)^A081464(n))), where fract(x) = x-floor(x).

Extensions

a(16)-a(23) from Robert Price, May 09 2012

A153666 Greatest number m such that the fractional part of (3/2)^A153662(n) <= 1/m.

Original entry on oeis.org

2, 4, 16, 11, 16799, 11199, 5536, 92694, 61796, 41197, 23242, 55710, 137921, 257825, 5271294, 706641581, 471094387, 314062925
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Examples

			a(3)=16 since 1/17<fract((3/2)^A153662(3))=fract((3/2)^4)=0.0625=1/16.

Crossrefs

Cf. A002379, A081464, A153662, A153663, A153664, A153665, A153667, A153668.

Programs

  • Mathematica
    A153662 = {1, 2, 4, 7, 3328, 3329, 4097, 12429, 12430, 12431, 18587, 44257, 112896, 129638, 4264691, 144941960, 144941961, 144941962};
Table[fp = FractionalPart[(3/2)^A153662[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++];  m - 1, {n, 1, Length[A153662]}] (* Robert Price, Mar 26 2019 *)

Formula

a(n):=floor(1/fract((3/2)^A153662(n))), where fract(x) = x-floor(x).

Extensions

a(15)-a(18) from Robert Price, May 09 2012
Showing 1-10 of 21 results. Next

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