A153710
Numbers k such that the fractional part of Pi^k is less than 1/k.
Original entry on oeis.org
1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479, 12128, 65875, 118885
Offset: 1
a(4) = 9 since fract(Pi^9) = 0.0993... < 1/9, but fract(Pi^k) = 0.3891..., 0.2932..., 0.5310... for 6 <= k <= 8, which all are greater than 1/k.
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Select[Range[1000], N[FractionalPart[Pi^#], 100] < (1/#) &] (* G. C. Greubel, Aug 25 2016 *)
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isok(k) = frac(Pi^k) < 1/k; \\ Michel Marcus, Feb 11 2014
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
a(5)=3, since 1-(1/4)=0.75>fract((3/2)^A153663(5))=fract((3/2)^12)=0.746...>=1-(1/3).
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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 *)
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
a(2)=14, since 1-(1/15)=0.933...>fract((3/2)^A153664(2))=fract((3/2)^14)=0.929...>=1-(1/14).
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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 *)
A153714
Greatest number m such that the fractional part of Pi^A153710(n) <= 1/m.
Original entry on oeis.org
7, 159, 50, 10, 21, 55, 117, 270, 307, 744, 757, 7804, 13876, 62099, 70718, 154755
Offset: 1
a(2)=159 since 1/160<fract(Pi^A153710(2))=fract(Pi^3)=0.0062766...<=1/159.
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A153710 = {1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479,
12128, 65875, 118885};
Table[fp = FractionalPart[Pi^A153710[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153710]}] (* Robert Price, May 10 2019 *)
A153718
Numbers k such that the fractional part of (Pi-2)^k is less than 1/k.
Original entry on oeis.org
1, 2, 23, 24, 35, 41, 65, 182, 72506, 107346
Offset: 1
a(3)=23 since fract((Pi-2)^23) = 0.0260069... < 1/23, but fract((Pi-2)^k) >= 1/k for 3 <= k <= 22.
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Select[Range[1000], N[FractionalPart[(Pi - 2)^#], 100] < (1/#) &] (* G. C. Greubel, Aug 25 2016 *)
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lista(nn) = for (n=1, nn, default(realprecision, n); if (frac((Pi-2)^n) < 1/n, print1(n, ", "))); \\ Michel Marcus, Nov 16 2014
A153722
Greatest number m such that the fractional part of (Pi-2)^A153718(n) <= 1/m.
Original entry on oeis.org
7, 3, 38, 318, 78, 83, 265, 185, 73351, 356362
Offset: 1
a(3) = 38 since 1/39 < fract((Pi-2)^A153718(3)) = fract((Pi-2)^23) = 0.02600... <= 1/38.
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A153718 = {1, 2, 23, 24, 35, 41, 65, 182, 72506, 107346};
Table[Floor[1/FractionalPart[(Pi - 2)^A153718[[n]]]], {n, 1,
Length[A153718]}] (* Robert Price, May 10 2019 *)
A153706
Greatest number m such that the fractional part of e^A153702(n) <= 1/m.
Original entry on oeis.org
1, 2, 11, 11, 964, 34015, 156075, 952945, 170942, 247768, 397506
Offset: 1
a(3) = 11 since 1/12 < fract(e^A153702(3)) = fract(e^3) = 0.0855... <= 1/11.
A153698
Greatest number m such that the fractional part of (10/9)^A153694(n) <= 1/m.
Original entry on oeis.org
9, 4, 11, 82, 6131, 4549, 26735, 8620, 14923, 20328, 151439, 227416, 771341, 2712159, 2676962, 2409266, 4490404, 4041364
Offset: 1
a(3) = 11 since 1/12 < fract((10/9)^A153694(3)) = fract((10/9)^7) = 0.09075... <= 1/11.
A153682
Greatest number m such that the fractional part of (1024/1000)^A153678(n) <= 1/m.
Original entry on oeis.org
41, 20, 13, 10, 7, 6, 718, 1350, 12472, 811799, 11462221, 8698270, 56414953
Offset: 1
a(5) = 7 since 1/8 < fract((1024/1000)^A153678(5)) = fract((1024/1000)^5) = 0.12589... <= 1/7.
A153674
Greatest number m such that the fractional part of (101/100)^A153670(n) <= 1/m.
Original entry on oeis.org
100, 49, 33, 24, 19, 16, 13, 12, 10, 147, 703, 676, 932, 3389, 7089, 1129226, 1741049, 1356464, 1960780, 11014240, 75249086, 28657625, 132665447, 499298451
Offset: 1
a(5) = 19 since 1/20 < fract((101/100)^A153670(5)) = fract((101/100)^5) = 0.0510... <= 1/19.
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