A153670 -id:A153670 - cp's OEIS frontend

A153674 Greatest number m such that the fractional part of (101/100)^A153670(n) <= 1/m.

100, 49, 33, 24, 19, 16, 13, 12, 10, 147, 703, 676, 932, 3389, 7089, 1129226, 1741049, 1356464, 1960780, 11014240, 75249086, 28657625, 132665447, 499298451

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(5) = 19 since 1/20 < fract((101/100)^A153670(5)) = fract((101/100)^5) = 0.0510... <= 1/19.

Cf. A153662, A154130, A153666, A153682, A153690, A153698, A153706, A153714, A153722.

a(n) = floor(1/fract((101/100)^A153670(n))), where fract(x) = x - floor(x).

a(18)-a(24) from Jinyuan Wang, Mar 03 2020

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

A153714 Greatest number m such that the fractional part of Pi^A153710(n) <= 1/m.

Original entry on oeis.org

7, 159, 50, 10, 21, 55, 117, 270, 307, 744, 757, 7804, 13876, 62099, 70718, 154755

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(2)=159 since 1/160<fract(Pi^A153710(2))=fract(Pi^3)=0.0062766...<=1/159.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A154130, A153722.

Cf. A001672.

A153710 = {1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479,
   12128, 65875, 118885};
Table[fp = FractionalPart[Pi^A153710[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153710]}] (* Robert Price, May 10 2019 *)

a(n) = floor(1/fract(Pi^A153710(n))), where fract(x) = x-floor(x).

a(16) from Robert Price, May 10 2019

A153718 Numbers k such that the fractional part of (Pi-2)^k is less than 1/k.

Original entry on oeis.org

1, 2, 23, 24, 35, 41, 65, 182, 72506, 107346

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract((Pi-2)^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 200000.
a(11) > 10^6. - Jon E. Schoenfield, Nov 15 2014

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

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A153722, A154130.

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

lista(nn) = for (n=1, nn, default(realprecision, n); if (frac((Pi-2)^n) < 1/n, print1(n, ", "))); \\ Michel Marcus, Nov 16 2014

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

Original entry on oeis.org

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).

A153706 Greatest number m such that the fractional part of e^A153702(n) <= 1/m.

Original entry on oeis.org

1, 2, 11, 11, 964, 34015, 156075, 952945, 170942, 247768, 397506

Offset: 1

Hieronymus Fischer, Jan 06 2009

			a(3) = 11 since 1/12 < fract(e^A153702(3)) = fract(e^3) = 0.0855... <= 1/11.

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A154130, A153714, A153722.

Cf. A000149.

Floor[1/(#-Floor[#])]&/@Exp[Select[Range[1000],FractionalPart[E^#]<(1/#)&]] (* Julien Kluge, Sep 20 2016 *)

a(n) = floor(1/fract(e^A153702(n))), where fract(x) = x - floor(x).

a(10)-a(11) from Jinyuan Wang, Mar 03 2020

cp's OEIS Frontend

A153674 Greatest number m such that the fractional part of (101/100)^A153670(n) <= 1/m.

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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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A153714 Greatest number m such that the fractional part of Pi^A153710(n) <= 1/m.

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A153718 Numbers k such that the fractional part of (Pi-2)^k is less than 1/k.

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