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 1-8 of 8 results.

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

A153664 Numbers k such that the fractional part of (3/2)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 14, 163, 1256, 2677, 8093, 49304, 49305, 158643, 164000, 835999, 2242294, 2242295, 2242296, 3965133, 25380333, 92600006, 92600007, 92600008, 92600009, 92600010, 92600011, 99267816, 125040717, 125040718
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2008

Keywords

Comments

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

Examples

			a(2) = 14 since fract((3/2)^14) = 0.92926... > 0.92857... = 1 - (1/14), but fract((3/2)^k) <= 1 - (1/k) for 1<k<14.

Crossrefs

Cf. A002379, A081464, A153662, A153663, A153665, A153666, A153667, A153668.

Programs

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

Extensions

a(11)-a(25) from Robert Gerbicz, Nov 21 2010

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

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

A153675 Greatest number m such that the fractional part of (101/100)^A153671(m) >= 1-(1/m).

Original entry on oeis.org

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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 9, 11, 13, 19, 30, 76, 81, 238, 913, 1334, 4645, 6812, 17396, 351085, 552184
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(5)=1, since 1-(1/2)=0.5>fract((101/100)^A153671(5))=fract((101/100)^5)=0.0510...>=1-(1/1).

Crossrefs

Cf. A153663, A154130, A153667, A153683, A153691, A153699, A153707, A153715, A153723.

Formula

a(n):=floor(1/(1-fract((101/100)^A153671(n)))), where fract(x) = x-floor(x).

A154136 Greatest number m such that the fractional part of (4/3)^A154132(m) >= 1-(1/m).

Original entry on oeis.org

1, 4, 88, 1228, 2253, 4562, 8183, 167378
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Examples

			a(3)=88, since 1-(1/89)=0.988764...>fract((4/3)^A154132(3))=fract((4/3)^8)=0.988721...>0.988636...=1-(1/88).

Crossrefs

Cf. A153667, A153675, A153707, A153715, A154144, A154152.
Cf. A002379, A064628.

Formula

a(n):=floor(1/(1-fract((4/3)^A154132(n)))), where fract(x) = x-floor(x).
Showing 1-8 of 8 results.

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