A153718 -id:A153718 - cp's OEIS frontend

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

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).

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

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

A153686 Numbers k such that the fractional part of (11/10)^k is less than 1/k.

Original entry on oeis.org

1, 2, 3, 17, 37, 48, 237, 420, 599, 615, 6638, 13885, 13886, 62963, 1063942, 9479731

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract((11/10)^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 2*10^5.
a(17) > 10^7. - Robert Price, Mar 19 2019

			a(4) = 17 since fract((11/10)^17) = 0.05447... < 1/17, but fract((11/10)^k) >= 1/k for 4 <= k <= 16.

Cf. A153662, A153670, A153682, A153678, A154130, A153694, A153702, A153710, A153718.

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

A153686_list, k, k10, k11 =  [], 1, 10, 11
while k < 10**6:
    if (k11 % k10)*k < k10:
        A153686_list.append(k)
    k += 1
    k10 *= 10
    k11 *= 11 # Chai Wah Wu, Apr 01 2021

a(15)-a(16) from Robert Price, Mar 19 2019

A153694 Numbers k such that the fractional part of (10/9)^k is less than 1/k.

Original entry on oeis.org

1, 2, 7, 62, 324, 1647, 3566, 5464, 8655, 8817, 123956, 132891, 182098, 566593, 2189647, 2189648, 3501843, 3501844

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract((10/9)^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 2*10^5.
a(19) > 10^7. - Robert Price, Mar 24 2019
Given a number k that is not only a term of this sequence but also has the property that the integer part of (10/9)^k is divisible by 9, we can expect that k+1 will likely also be a term of the sequence. E.g., k = 2189647 is a term because fract((10/9)^k) = 0.000000373557... < 0.000000456694... = 1/k, and since floor((10/9)^k) is divisible by 9, the integer and fractional parts of (10/9)^(k+1) will be exactly 10/9 times the integer and fractional parts of (10/9)^k, respectively, yielding a fractional part (10/9) * 0.000000373557... = 0.000000415064... < 0.000000456694... = 1/(k+1), so k+1 = 2189648 is also a term. - Jon E. Schoenfield, Mar 24 2019

			a(3) = 7 since fract((10/9)^7) = 0.09075... < 1/7, but fract((10/9)^k) >= 1/k for 3 <= k <= 6.

Cf. A153662, A153670, A153678, A153686, A153698, A154130, A153702, A153710, A153718.

Select[Range[1000], FractionalPart[(10/9)^#] < (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

a(14)-a(18) from Robert Price, Mar 24 2019

A153702 Numbers k such that the fractional part of e^k is less than 1/k.

Original entry on oeis.org

1, 2, 3, 9, 732, 5469, 28414, 37373, 93638, 136986, 192897

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract(e^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(12) > 300000. - Robert Price, Mar 23 2019

			a(4) = 9 since fract(e^9) = 0.08392... < 1/9, but fract(e^k) = 0.598..., 0.413..., 0.428..., 0.633..., 0.957... for 4 <= k <= 8, which are all greater than 1/k.

Cf. A153662, A153670, A153678, A153686, A153694, A153706, A154130, A153710, A153718.

Cf. A000149.

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

a(10)-a(11) 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

cp's OEIS Frontend

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

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

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A153686 Numbers k such that the fractional part of (11/10)^k is less than 1/k.

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A153694 Numbers k such that the fractional part of (10/9)^k is less than 1/k.

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A153702 Numbers k such that the fractional part of e^k is less than 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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