A154130 -id:A154130 - cp's OEIS frontend

A153671 Minimal exponents m such that the fractional part of (101/100)^m obtains a maximum (when starting with m=1).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 110, 180, 783, 859, 1803, 7591, 10763, 19105, 50172, 355146, 1101696, 1452050, 3047334, 3933030

Offset: 1

Hieronymus Fischer, Jan 06 2009

nonn

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of (101/100)^m is greater than the fractional part of (101/100)^k for all k, 1<=k

The next such number must be greater than 10^6.

a(84) > 10^7. Robert Price, Mar 21 2019

			a(5)=5, since fract((101/100)^5)=0.05101005, but fract((101/100)^k)=0.01, 0.0201, 0.030301, 0.04060401 for 1<=k<=4; thus fract((101/100)^5)>fract((101/100)^k) for 1<=k<5.

Cf. A153663, A154130, A153675, A153679, A153687, A153695, A091560, A153711, A153719.

p = 0; Select[Range[1, 20000],
If[FractionalPart[(101/100)^#] > p, p = FractionalPart[(101/100)^#];
True] &] (* Robert Price, Mar 21 2019 *)

A153671_list, m, n, k, q = [], 1, 101, 100, 0
while m < 10**4:
    r = n % k
    if r > q:
        q = r
        A153671_list.append(m)
    m += 1
    n *= 101
    k *= 100
    q *= 100 # Chai Wah Wu, May 16 2020

Recursion: a(1):=1, a(k):=min{ m>1 | fract((101/100)^m) > fract((101/100)^a(k-1))}, where fract(x) = x-floor(x).

a(72)-a(83) from Robert Price, Mar 21 2019

A153669 Minimal exponents m such that the fractional part of (101/100)^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 70, 209, 378, 1653, 2697, 4806, 13744, 66919, 67873, 75666, 81125, 173389, 529938, 1572706, 4751419, 7159431, 7840546, 15896994, 71074288, 119325567

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 (101/100)^m is less than the fractional part of (101/100)^k for all k, 1<=k

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

			a(2)=70, since fract((101/100)^70)=0.00676..., but fract((101/100)^k)>=0.01 for 1<=k<=69; thus fract((101/100)^70)<fract((101/100)^k) for 1<=k<70.

Cf. A081464, A154130, A153673, A153677, A153685, A153693, A153701, A137994, A153717.

p = 1; Select[Range[1, 5000],
If[FractionalPart[(101/100)^#] < p, p = FractionalPart[(101/100)^#];
True] &] (* Robert Price, Mar 21 2019 *)

Recursion: a(1):=1, a(k):=min{ m>1 | fract((101/100)^m) < fract((101/100)^a(k-1))}, where fract(x) = x-floor(x).

a(15)-a(21) from Robert Gerbicz, Nov 22 2010

A153670 Numbers k such that the fractional part of (101/100)^k is less than 1/k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 70, 209, 241, 378, 2697, 4806, 173389, 529938, 1334508, 1572706, 7840546, 15896994, 20204295, 71074288, 119325567

Offset: 1

Hieronymus Fischer, Jan 06 2009

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

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

			a(10) = 70 since fract((101/100)^70) = 0.006... < 1/10, but fract((101/100)^k) > 0.1 >= 1/k for 10 <= k <= 69.

Cf. A153662, A154130, A153674, A153678, A153686, A153694, A153702, A153710, A153718.

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

from itertools import count, islice
def A153670gen(): # generator of terms
    k10, k11 = 100, 101
    for k in count(1):
        if (k11 % k10)*k < k10:
            yield k
        k10 *= 100
        k11 *= 101
A153670_list = list(islice(A153670gen(),16)) # Chai Wah Wu, Dec 23 2021

a(18)-a(24) from Robert Gerbicz, Nov 29 2010

A153672 Numbers k such that the fractional part of (101/100)^k is greater than 1-(1/k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Jan 06 2009

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.

			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.

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

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

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

A137994 a(n) is the smallest integer > a(n-1) such that {Pi^a(n)} < {Pi^a(n-1)}, where {x} = x - floor(x), a(1)=1.

Original entry on oeis.org

1, 3, 81, 264, 281, 472, 1147, 2081, 3207, 3592, 10479, 12128, 65875, 114791, 118885

Offset: 1

Leroy Quet and M. F. Hasler, Mar 14 2008

The sequence was suggested by Leroy Quet on Pi day 2008, cf. A138324.
The next such number must be greater than 100000. - Hieronymus Fischer, Jan 06 2009
a(16) > 300,000. - Robert Price, Mar 25 2019

			a(3)=81, since {Pi^81}=0.0037011283.., but {Pi^k}>=0.0062766802... for 1<=k<=80; thus {Pi^81}<{Pi^k} for 1<=k<81. - _Hieronymus Fischer_, Jan 06 2009

Cf. A001203, A138324, A001672.

Cf. A081464, A153669, A153677, A153685, A153693, A153705, A153713, A154130, A153717. - Hieronymus Fischer, Jan 06 2009

$MaxExtraPrecision = 10000;
p = .999;
Select[Range[1, 5000],
If[FractionalPart[Pi^#] < p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 12 2019 *)

default(realprecision,10^4); print1(a=1); for(i=1,100, f=frac(Pi^a); until( frac(Pi^a++)

a(11)-a(13) from Hieronymus Fischer, Jan 06 2009
Edited by R. J. Mathar, May 21 2010
a(14)-a(15) from Robert Price, Mar 12 2019

A153677 Minimal exponents m such that the fractional part of (1024/1000)^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 68, 142, 341, 395, 490, 585, 1164, 1707, 26366, 41358, 46074, 120805, 147332, 184259, 205661, 385710, 522271, 3418770, 3675376, 9424094

Offset: 1

Hieronymus Fischer, Jan 06 2009

Recursive definition: a(1)=1, a(n) is the least positive integer m such that the fractional part of (1024/1000)^m is less than the fractional part of (1024/1000)^k for all k, 1 <= k < m.
a(21) >= 4.5*10^6. - David A. Corneth, Mar 15 2019
a(22) > 10^7. Robert Price, Mar 16 2019

			a(2)=68, since fract((1024/1000)^68) = 0.016456..., but fract((1024/1000)^k) >= 0.024 for 1 <= k <= 67; thus fract((1024/1000)^68) < fract((1024/1000)^k) for 1 <= k < 68.

Cf. A081464, A153669, A153681, A154130, A153685, A153693, A153701, A137994, A153717.

$MaxExtraPrecision = 10000;
p = .999;
Select[Range[1, 50000],
If[FractionalPart[(1024/1000)^#] < p,
p = FractionalPart[(1024/1000)^#]; True] &] (* Robert Price, Mar 15 2019 *)

upto(n) = my(res = List(), r = 1, p = 1); for(i=1, n, c = frac(p *= 1.024); if(cDavid A. Corneth, Mar 15 2019

Recursion: a(1):=1, a(k):=min{ m>1 | fract((1024/1000)^m) < fract((1024/1000)^a(k-1))}, where fract(x) = x-floor(x).

a(18) from Robert Price, Mar 15 2019
a(19)-a(20) from David A. Corneth, Mar 15 2019
a(21) from Robert Price, Mar 16 2019

A153678 Numbers k such that the fractional part of (1024/1000)^k is less than 1/k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 585, 1164, 1707, 522271, 3675376, 3906074, 9424094

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract((1024/1000)^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 5*10^5.
a(14) > 10^7. - Robert Price, Mar 16 2019

			a(7) = 585 since fract((1024/1000)^585) = 0.00139... < 1/585, but fract((1024/1000)^k) >= 1/k for 7 <= k <= 584.

Cf. A153662, A153670, A153682, A154130, A153686, A153694, A153702, A153710, A153718.

Select[Range[2000], FractionalPart[(1024/1000)^#] < (1/#) &] (* G. C. Greubel, Aug 24 2016; corrected by Robert Price, Mar 16 2019 *)

isok(n) = frac((1024/1000)^n) < 1/n \\ Michel Marcus, Aug 06 2013

a(10)-a(13) from Robert Price, Mar 16 2019

A153679 Minimal exponents m such that the fractional part of (1024/1000)^m obtains a maximum (when starting with m=1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 82, 134, 1306, 2036, 6393, 34477, 145984, 2746739, 2792428, 8460321

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 (1024/1000)^m is greater than the fractional part of (1024/1000)^k for all k, 1<=k

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

a(40) > 10^7. Robert Price, Mar 16 2019

			a(30)=82, since fract((1024/1000)^82)= 0.99191990..., but fract((1024/1000)^k)<0.9893 for 1<=k<=81; thus fract((1024/1000)^82)>fract((1024/1000)^k) for 1<=k<82 and 82 is the minimal exponent >29 with this property.

Cf. A153663, A153671, A153675, A154130, A153687, A153695, A091560, A153711, A153719.

$MaxExtraPrecision = 10000;
p = 0;
Select[Range[1, 50000],
If[FractionalPart[(1024/1000)^#] > p,
p = FractionalPart[(1024/1000)^#]; True] &] (* Robert Price, Mar 16 2019 *)

A153679_list, m, n, k, q = [], 1, 1024, 1000, 0
while m < 10**4:
    r = n % k
    if r > q:
        q = r
        A153679_list.append(m)
    m += 1
    n *= 1024
    k *= 1000
    q *= 1000 # Chai Wah Wu, May 16 2020

Recursion: a(1):=1, a(k):=min{ m>1 | fract((1024/1000)^m) > fract((1024/1000)^a(k-1))}, where fract(x) = x-floor(x).

a(37)-a(39) from Robert Price, Mar 16 2019

A153680 Numbers k such that the fractional part of (1024/1000)^k is greater than 1-(1/k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Jan 06 2009

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.

			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.

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

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

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

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cp's OEIS Frontend

A154134 Greatest number m such that the fractional part of (4/3)^A154130(n) <= 1/m.

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A153671 Minimal exponents m such that the fractional part of (101/100)^m obtains a maximum (when starting with m=1).

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A153669 Minimal exponents m such that the fractional part of (101/100)^m obtains a minimum (when starting with m=1).

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A153670 Numbers k such that the fractional part of (101/100)^k is less than 1/k.

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A153672 Numbers k such that the fractional part of (101/100)^k is greater than 1-(1/k).

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A137994 a(n) is the smallest integer > a(n-1) such that {Pi^a(n)} < {Pi^a(n-1)}, where {x} = x - floor(x), a(1)=1.

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A153677 Minimal exponents m such that the fractional part of (1024/1000)^m obtains a minimum (when starting with m=1).

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A153678 Numbers k such that the fractional part of (1024/1000)^k is less than 1/k.

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