A154130 -id:A154130 - cp's OEIS frontend

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

3, 23, 27, 261, 348, 2720, 72944, 347065, 244543

Offset: 1

Hieronymus Fischer, Jan 06 2009

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

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

Cf. A000149.

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 *)

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

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

Hieronymus Fischer, Jan 06 2009

			a(2)=159 since 1/160<fract(Pi^A153710(2))=fract(Pi^3)=0.0062766...<=1/159.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A154130, A153722.

Cf. A001672.

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 *)

a(n) = floor(1/fract(Pi^A153710(n))), where fract(x) = x-floor(x).

a(16) 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

Hieronymus Fischer, Jan 06 2009

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

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

Cf. A001672.

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 *)

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

a(14) from 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

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract((Pi-2)^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 200000.
a(11) > 10^6. - Jon E. Schoenfield, Nov 15 2014

			a(3)=23 since fract((Pi-2)^23) = 0.0260069... < 1/23, but fract((Pi-2)^k) >= 1/k for 3 <= k <= 22.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A153722, A154130.

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

lista(nn) = for (n=1, nn, default(realprecision, n); if (frac((Pi-2)^n) < 1/n, print1(n, ", "))); \\ Michel Marcus, Nov 16 2014

A153719 Minimal exponents m such that the fractional part of (Pi-2)^m obtains a maximum (when starting with m=1).

Original entry on oeis.org

1, 2, 3, 4, 5, 39, 56, 85, 557, 911, 2919, 2921, 4491, 11543, 15724, 98040, 110932, 126659

Offset: 1

Hieronymus Fischer, Jan 06 2009

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of (Pi-2)^m is greater than the fractional part of (Pi-2)^k for all k, 1<=k

The next such number must be greater than 200000.

a(19) > 300000. - Robert Price, Mar 26 2019

			a(6)=39, since fract((Pi-2)^39)= 0.9586616565..., but fract((Pi-2)^k)<=0.9389018... for 1<=k<=38; thus fract((Pi-2)^39)>fract((Pi-2)^k) for 1<=k<39 and 39 is the minimal exponent > 5 with this property.

Cf. A153663, A153671, A153679, A153687, A153695, A153707, A153715, A153723, A154130.

$MaxExtraPrecision = 100000;
p = 0; Select[Range[1, 10000],
If[FractionalPart[(Pi - 2)^#] > p, p = FractionalPart[(Pi - 2)^#];
True] &] (* Robert Price, Mar 26 2019 *)

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

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

Hieronymus Fischer, Jan 06 2009

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

			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.

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

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

A153721 Greatest number m such that the fractional part of (Pi-2)^A153717(n) <= 1/m.

Original entry on oeis.org

7, 7, 38, 318, 393, 396, 484, 2076, 2619, 4099, 5264, 8556, 18070, 20732, 27209, 73351, 356362

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(3)=38 since 1/39<fract((Pi-2)^A153717(3))=fract((Pi-2)^23)=0.02600...<=1/38.

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A137994, A153717, A154130.

A153717 = {1, 20, 23, 24, 523, 2811, 3465, 3776, 4567, 6145, 8507, 9353, 19790, 41136, 62097, 72506, 107346};
Table[fp = FractionalPart[(Pi - 2)^A153717[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153717]}] (* Robert Price, Mar 26 2019 *)

a(n) = floor(1/fract((Pi-2)^A153717(n))), where fract(x) = x-floor(x).

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

Hieronymus Fischer, Jan 06 2009

			a(3) = 38 since 1/39 < fract((Pi-2)^A153718(3)) = fract((Pi-2)^23) = 0.02600... <= 1/38.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A153718, A154130.

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 *)

a(n) = floor(1/fract((Pi-2)^A153718(n))), where fract(x) = x-floor(x).

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

Hieronymus Fischer, Jan 06 2009

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

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

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 *)

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

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

Hieronymus Fischer, Jan 06 2009

			a(3) = 11 since 1/12 < fract(e^A153702(3)) = fract(e^3) = 0.0855... <= 1/11.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A154130, A153714, A153722.

Cf. A000149.

Floor[1/(#-Floor[#])]&/@Exp[Select[Range[1000],FractionalPart[E^#]<(1/#)&]] (* Julien Kluge, Sep 20 2016 *)

a(n) = floor(1/fract(e^A153702(n))), where fract(x) = x - floor(x).

a(10)-a(11) from Jinyuan Wang, Mar 03 2020

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

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A153714 Greatest number m such that the fractional part of Pi^A153710(n) <= 1/m.

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

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A153718 Numbers k such that the fractional part of (Pi-2)^k is less than 1/k.

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A153719 Minimal exponents m such that the fractional part of (Pi-2)^m obtains a maximum (when starting with m=1).

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

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A153721 Greatest number m such that the fractional part of (Pi-2)^A153717(n) <= 1/m.

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A153722 Greatest number m such that the fractional part of (Pi-2)^A153718(n) <= 1/m.

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

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