A154130 -id:A154130 - cp's OEIS frontend

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

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

Offset: 1

Hieronymus Fischer, Jan 06 2009

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

			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).

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

Cf. A000149.

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

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

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

Hieronymus Fischer, Jan 08 2009

Numbers k such that fract(Pi^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(17) > 300000. - Robert Price, Mar 25 2019

			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.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153714, A154130, A153718.

Cf. A001672.

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

isok(k) = frac(Pi^k) < 1/k; \\ Michel Marcus, Feb 11 2014

a(16) from Robert Price, Mar 25 2019

A153712 Numbers k such that the fractional part of Pi^k is greater than 1-(1/k).

Original entry on oeis.org

1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322, 5269, 151710

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract(Pi^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(15) > 300000. - Robert Price, Mar 25 2019

			a(3) = 15, since fract(Pi^15) = 0.969... > 0.933... = 1 - (1/15), but fract(Pi^k) <= 1 - (1/k) for 3 <= k <= 14.

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153716, A154130, A153720.

Cf. A001672.

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

a(14) from Robert Price, Mar 25 2019

A091560 Fractional part of e^a(n) is the largest yet.

Original entry on oeis.org

1, 8, 19, 76, 166, 178, 209, 1907, 20926, 22925, 32653, 119136

Offset: 1

Jon Perry, Mar 04 2004

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

The next such number must be greater than 100000. [Hieronymus Fischer, Jan 06 2009]

a(13) > 300,000. Robert Price, Mar 23 2019

			a(2)=8, since fract(e^8)= 0.9579870417..., but fract(e^k)<=0.7182818... for 1<=k<=7;
thus fract(e^8)>fract(e^k) for 1<=k<8 and 8 is the minimal exponent > 1 with this property. [_Hieronymus Fischer_, Jan 06 2009]

Cf. A153663, A153671, A153679, A153687, A153695, A153707, A154130, A153711, A153719, A000149. [Hieronymus Fischer, Jan 06 2009]

a = 0; Do[b = N[ FractionalPart[ N[ E, 2^12]^n], 24]; If[b > a, Print[n]; a = b], {n, 1, 9400}] (* Robert G. Wilson v, Mar 16 2004 *)

E=exp(1); /* use sufficient precision! */
ym=0;for(i=1,1000,x=E^i;y=x-floor(x);if(y>ym,print1(","i);ym=y))

Recursion: a(1):=1, a(k):=min{ m>1 | fract(e^m) > fract(e^a(k-1))}, where fract(x) = x-floor(x). [Hieronymus Fischer, Jan 06 2009]

a(8) from Robert G. Wilson v, Mar 16 2004
a(9)-a(11) from Hieronymus Fischer, Jan 06 2009
a(12) from Robert Price, Mar 23 2019

A153707 Greatest number m such that the fractional part of e^A091560(m) >= 1-(1/m).

Original entry on oeis.org

3, 23, 27, 41, 59, 261, 348, 2720, 3198, 6064, 72944, 347065

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(2)=23, since 1-(1/24) = 0.9583...> fract(e^A091560(2)) = fract(e^8) = 0.95798.. >= 0.95652... >= 1-(1/23).

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

Cf. A000149.

$MaxExtraPrecision = 100000;
A091560 = {1,8,19,76,166,178,209,1907,20926,22925,32653,119136};
Floor[1/(1-FractionalPart[E^A091560])] (* Robert Price, Apr 18 2019 *)

a(n):=floor(1/(1-fract(e^A091560(n)))), where fract(x) = x-floor(x).

a(12) from Robert Price, Apr 18 2019

A153711 Minimal exponents m such that the fractional part of Pi^m obtains a maximum (when starting with m=1).

Original entry on oeis.org

1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710

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^m is greater than the fractional part of Pi^k for all k, 1<=k

The next such number must be greater than 100000.

a(11) > 300000. - Robert Price, Mar 25 2019

			a(3)=15, since fract(Pi^15)= 0.9693879984..., but fract(Pi^k)<=0.8696... for 1<=k<=14; thus fract(Pi^15)>fract(Pi^k) for 1<=k<15 and 15 is the minimal exponent > 2 with this property.

Cf. A153663, A153671, A153679, A153687, A153695, A153707, A153715, A154130, A153719.

Cf. A001672.

$MaxExtraPrecision = 100000;
p = 0; Select[Range[1, 10000],
If[FractionalPart[Pi^#] > p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 25 2019 *)

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

a(9)-a(10) from Robert Price, Mar 25 2019

A153715 Greatest number m such that the fractional part of Pi^A153711(m) >= 1-(1/m).

Original entry on oeis.org

1, 7, 32, 53, 189, 2665, 10810, 26577, 128778, 483367

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153711(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).

Cf. A153663, A153671, A153679, A153687, A153695, A091560, A153711, A154130, A153723.

Cf. A001672.

$MaxExtraPrecision = 100000;
A153711 = {1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710};
Floor[1/(1-FractionalPart[Pi^A153711])] (* Robert Price, Apr 18 2019 *)

a(n) = floor(1/(1-fract(Pi^A153711(n)))), where fract(x) = x-floor(x).

a(9)-a(10) from Robert Price, Apr 18 2019

A153717 Minimal exponents m such that the fractional part of (Pi-2)^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 20, 23, 24, 523, 2811, 3465, 3776, 4567, 6145, 8507, 9353, 19790, 41136, 62097, 72506, 107346

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 less than the fractional part of (Pi-2)^k for all k, 1<=k

The next such number must be greater than 200000.

a(18) > 300000. - Robert Price, Mar 26 2019

			a(3)=23, since fract((Pi-2)^23)=0.0260069.., but fract((Pi-2)^k)>=0.1326... for 1<=k<=22; thus fract((Pi-2)^23)<fract((Pi-2)^k) for 1<=k<23.

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A137994, A153721, A154130.

$MaxExtraPrecision = 100000;
p = 1; 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).

A153723 Greatest number m such that the fractional part of (Pi-2)^A153719(m) >= 1-(1/m).

Original entry on oeis.org

1, 1, 1, 3, 16, 24, 45, 158, 410, 946, 1182, 8786, 16159, 20188, 61392, 78800, 78959, 217556

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(5) = 16, since 1-(1/17) = 0.941176... > fract((Pi-2)^A153719(5)) = fract((Pi-2)^5) = 0.9389... >= 0.9375 = 1-(1/16).

Cf. A153663, A153671, A153679, A153687, A153695, A091560, A153711, A153719, A154130.

$MaxExtraPrecision = 100000;
A153719 = {1, 2, 3, 4, 5, 39, 56, 85, 557, 911, 2919, 2921, 4491,
   11543, 15724, 98040, 110932, 126659};
Floor[1/(1 - FractionalPart[(Pi - 2)^A153719])] (* Robert Price, Apr 18 2019 *)

a(n) = floor(1/(1-fract((Pi-2)^A153719(n)))), where fract(x) = x-floor(x).

A153705 Greatest number m such that the fractional part of e^A153701(n) <= 1/m.

Original entry on oeis.org

1, 2, 11, 11, 23, 28, 69, 85, 115, 964, 1153, 1292, 1296, 1877, 34015, 156075, 952945

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(3)=11 since 1/12 < fract(e^A153701(3)) = fract(e^3) = 0.0855... <= 1/11.

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A154130, A153713, A153721.

Cf. A000149.

A153701 = {1, 2, 3, 9, 29, 45, 75, 135, 219, 732, 1351, 3315, 4795,
   4920, 5469, 28414, 37373};
Table[fp = FractionalPart[E^A153701[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153701]}] (* Robert Price, Mar 25 2019 *)

a(n) = floor(1/fract(e^A153701(n))), where fract(x) = x-floor(x).

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

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A153710 Numbers k such that the fractional part of Pi^k is less than 1/k.

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A153712 Numbers k such that the fractional part of Pi^k is greater than 1-(1/k).

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A091560 Fractional part of e^a(n) is the largest yet.

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A153707 Greatest number m such that the fractional part of e^A091560(m) >= 1-(1/m).

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A153711 Minimal exponents m such that the fractional part of Pi^m obtains a maximum (when starting with m=1).

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A153715 Greatest number m such that the fractional part of Pi^A153711(m) >= 1-(1/m).

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A153717 Minimal exponents m such that the fractional part of (Pi-2)^m obtains a minimum (when starting with m=1).

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