A153696 -id:A153696

A153700

Greatest number m such that the fractional part of (10/9)^A153696(n) >= 1-(1/m).

+20
4

1, 8, 15, 264, 8741, 15912, 409895

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..7.

FORMULA

a(n):=floor(1/(1-fract((10/9)^A153696(n)))), where fract(x) = x-floor(x).

EXAMPLE

a(3)=15, since 1-(1/16)=0.9375>fract((10/9)^A153696(3))=fract((10/9)^13)=0.9341...>=1-(1/15).

CROSSREFS

Cf. A153664, A153672, A153680, A153688, A153696, A154130, A153708, A153716, A153724.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

STATUS

approved

A153672

Numbers k such that the fractional part of (101/100)^k is greater than 1-(1/k).

+10
14

1, 69, 180, 783, 859, 1803, 10763, 19105, 39568, 50172, 132572, 355146, 1452050, 2245950, 3047334, 3933030, 4165171, 98544173

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers k such that fract((101/100)^k) > 1-(1/k), where fract(x) = x-floor(x).

The next term is greater than 2*10^8.

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

a(2) = 69, since fract((101/100)^69) = 0.9868... > 0.9855... = 1 - (1/69), but fract((101/100)^k) <= 1 - (1/k) for 1 < k < 69.

MATHEMATICA

Select[Range[1000], FractionalPart[(101/100)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

CROSSREFS

Cf. A153664, A154130, A153676, A153680, A153688, A153696, A153704, A153712, A153720.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

a(13)-a(18) from Robert Gerbicz, Nov 22 2010

STATUS

approved

A153680

Numbers k such that the fractional part of (1024/1000)^k is greater than 1-(1/k).

+10
11

1, 29, 82, 134, 277, 1306, 2036, 2349, 6393, 9389, 9816, 21689, 34477, 145984, 171954, 956357, 2746739

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers k such that fract((1024/1000)^k) > 1-(1/k), where fract(x) = x-floor(x).

The next such number must be greater than 5*10^5.

LINKS

Table of n, a(n) for n=1..17.

EXAMPLE

a(2) = 29, since fract((1024/1000)^29) = 0.98929... > 0.9655... = 1 - (1/29), but fract((1024/1000)^k) <= 1 - (1/k) for 1 < k < 29.

MATHEMATICA

Select[Range[1000], FractionalPart[(1024/1000)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

CROSSREFS

Cf. A153664, A153672, A153684, A154130, A153688, A153696, A153704, A153712, A153720.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

a(16)-a(17) from Hagen von Eitzen, May 16 2009

STATUS

approved

A153688

Numbers k such that the fractional part of (11/10)^k is greater than 1-(1/k).

+10
11

1, 7, 77, 103, 320, 1821, 2992, 15290, 88651, 88652, 180168, 410498, 548816, 672732

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers k such that fract((11/10)^k) > 1-(1/k), where fract(x) = x-floor(x).

The next such number must be greater than 2*10^5.

a(15) > 10^7. Robert Price, Mar 19 2019

LINKS

Table of n, a(n) for n=1..14.

EXAMPLE

a(2) = 7, since fract((11/10)^7) = 0.9487... > 0.8571... = 1 - (1/7), but fract((11/10)^k) <= 1 - (1/k) for 1 < k < 7.

MATHEMATICA

Select[Range[1000], FractionalPart[(11/10)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

CROSSREFS

Cf. A153664, A153672, A153684, A153692, A154130, A153696, A153704, A153712, A153720.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

a(12)-a(14) from Robert Price, Mar 19 2019

STATUS

approved

A153704

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..9.

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

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

Cf. A000149.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

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

STATUS

approved

A153712

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..14.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

Cf. A001672.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

a(14) from Robert Price, Mar 25 2019

STATUS

approved

A153708

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..9.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

Cf. A000149.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

a(8)-a(9) from Robert Price, May 10 2019

STATUS

approved

A153716

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

+10
7

1, 7, 32, 53, 189, 131, 2665, 10810, 2693, 1976, 3697, 4289, 26577, 483367

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..14.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

Cf. A001672.

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

EXTENSIONS

a(14) from Robert Price, May 10 2019

STATUS

approved

A153720

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

+10
7

1, 5, 8, 85, 911, 2921, 4491, 11543, 15724, 27683, 29921, 37276, 126659

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..13.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

STATUS

approved

A153724

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

+10
7

1, 16, 8, 158, 946, 8786, 16159, 20188, 61392, 34039, 31425, 59154, 217556

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..13.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Hieronymus Fischer, Jan 06 2009

STATUS

approved