A153707 -id:A153707 - cp's OEIS frontend

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

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

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

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).

A153699 Greatest number m such that the fractional part of (10/9)^A153695(m) >= 1-(1/m).

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 15, 264, 334, 465, 683, 713, 758, 8741, 15912, 18920, 38560, 409895

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(7)=15, since 1-(1/16)=0.9375>fract((10/9)^A153695(7))=fract((10/9)^13)=0.9341...>=1-(1/15).

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

a(n):=floor(1/(1-fract((10/9)^A153695(n)))), where fract(x) = x-floor(x).

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

Hieronymus Fischer, Jan 06 2009

nonn

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

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

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

A153683 Greatest number m such that the fractional part of (1024/1000)^A153679(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, 2, 2, 2, 2, 3, 3, 4, 5, 6, 9, 17, 93, 123, 1061, 1360, 4137, 66910, 571666, 1192010

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(5)=1, since 1-(1/2)=0.5>fract((1024/1000)^A153679(5))=fract((1024/1000)^5)=0.0510...>=1-(1/1).

Cf. A153663, A153671, A153679, A154130, A153691, A153699, A153707, A153715, A153723.

a(n):=floor(1/(1-fract((1024/1000)^A153679(n)))), where fract(x) = x-floor(x).

A153691 Greatest number m such that the fractional part of (11/10)^A153687(m) >= 1-(1/m).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 19, 21, 28, 151, 200, 709, 767, 5727, 15908, 162819, 302991

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(6)=4, since 1-(1/5)=0.8>fract((11/10)^A153687(6))=fract((11/10)^6)=0.771...>=1-(1/4).

Cf. A153663, A153671, A153679, A153683, A154130, A153699, A153707, A153715, A153723.

a(n):=floor(1/(1-fract((11/10)^A153687(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

Hieronymus Fischer, Jan 11 2009

			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).

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

Cf. A002379, A064628.

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

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cp's OEIS Frontend

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

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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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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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A153699 Greatest number m such that the fractional part of (10/9)^A153695(m) >= 1-(1/m).

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

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A153683 Greatest number m such that the fractional part of (1024/1000)^A153679(m) >= 1-(1/m).

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A153691 Greatest number m such that the fractional part of (11/10)^A153687(m) >= 1-(1/m).

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

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