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

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

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

A154137 Greatest number m such that the fractional part of (4/3)^A154133(n) >= 1-(1/m).

Original entry on oeis.org

1, 4, 88, 1228, 187, 4562, 8183, 167378, 35419, 77421, 5593723, 3306511, 83205705, 22413581, 24296709, 35457806, 26593355, 19945016, 80184972, 389460601
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Examples

			a(3)=88, since 1-(1/89)=0.988764...>fract((4/3)^A154133(3))=fract((4/3)^8)=0.988721...>0.988636...=1-(1/88).

Crossrefs

Cf. A153668, A153676, A153708, A153716, A154145, A154153.
Cf. A002379, A064628.

Formula

a(n):=floor(1/(1-fract((4/3)^A154133(n)))), where fract(x) = x-floor(x).

Extensions

a(11)-a(20) from Robert Price, May 10 2012
Showing 1-8 of 8 results.

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