A154130 -id:A154130 - cp's OEIS frontend

A153685 Minimal exponents m such that the fractional part of (11/10)^m obtains a minimum (when starting with m=1).

1, 17, 37, 237, 599, 615, 6638, 13885, 1063942, 9479731

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 (11/10)^m is less than the fractional part of (11/10)^k for all k, 1<=k

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

a(11) > 10^7. Robert Price, Mar 19 2019

			a(2)=17, since fract((11/10)^17)=0.05447.., but fract((11/10)^k)>=0.1 for 1<=k<=16; thus fract((11/10)^17)<fract((11/10)^k) for 1<=k<17.

Cf. A081464, A153669, A153689, A153677, A154130, A153693, A153701, A137994, A153717.

p = 1; Select[Range[1, 50000],
 If[FractionalPart[(11/10)^#] < p, p = FractionalPart[(11/10)^#];
True] &] (* Robert Price, Mar 19 2019 *)

Recursion: a(1):=1, a(k):=min{ m>1 | fract((11/10)^m) < fract((11/10)^a(k-1))}, where fract(x) = x-floor(x).

a(9)-a(10) from Robert Price, Mar 19 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

A153687 Minimal exponents m such that the fractional part of (11/10)^m obtains a maximum (when starting with m=1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 23, 56, 77, 103, 320, 1477, 1821, 2992, 15290, 180168, 410498, 548816, 672732, 2601223

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 (11/10)^m is greater than the fractional part of (11/10)^k for all k, 1<=k

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

a(22) > 10^7. Robert Price, Mar 19 2019

			a(8)=23, since fract((11/10)^23)= 0.9543..., but fract((11/10)^k)<0.95 for 1<=k<=22;
thus fract((11/10)^23)>fract((11/10)^k) for 1<=k<23 and 23 is the minimal exponent > 7 with this property.

Cf. A153663, A153671, A153683, A153679, A154130, A153695, A091560, A153711, A153719.

p = 0; Select[Range[1, 50000],
If[FractionalPart[(11/10)^#] > p, p = FractionalPart[(11/10)^#];
True] &] (* Robert Price, Mar 19 2019 *)

A153687_list, m, n, k, q = [], 1, 11, 10, 0
while m < 10**4:
    r = n % k
    if r > q:
        q = r
        A153687_list.append(m)
    m += 1
    n *= 11
    k *= 10
    q *= 10 # Chai Wah Wu, May 16 2020

Recursion: a(1):=1, a(k):=min{ m>1 | fract((11/10)^m) > fract((11/10)^a(k-1))}, where fract(x) = x-floor(x).

a(18)-a(21) from Robert Price, Mar 19 2019

A153688 Numbers k such that the fractional part of (11/10)^k is greater than 1-(1/k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Jan 06 2009

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

			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.

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

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

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

A153693 Minimal exponents m such that the fractional part of (10/9)^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 7, 50, 62, 324, 3566, 66877, 108201, 123956, 132891, 182098, 566593, 3501843

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 (10/9)^m is less than the fractional part of (10/9)^k for all k, 1 <= k < m.
The next such number must be greater than 2*10^5.
a(14) > 10^7. - Robert Price, Mar 24 2019

			a(2)=7, since fract((10/9)^7) = 0.09075.., but fract((10/9)^k) >= 0.11... for 1 <= k <= 6; thus fract((10/9)^7) < fract((10/9)^k) for 1 <= k < 7.

Cf. A081464, A153669, A153677, A153685, A153697, A154130, A153701, A137994, A153717.

$MaxExtraPrecision = 100000;
p = 1; Select[Range[1, 10000],
If[FractionalPart[(10/9)^#] < p, p = FractionalPart[(10/9)^#];
True] &] (* Robert Price, Mar 24 2019 *)

Recursion: a(1):=1, a(k):=min{ m>1 | fract((10/9)^m) < fract((10/9)^a(k-1))}, where fract(x) = x-floor(x).

a(12)-a(13) from Robert Price, Mar 24 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

A153695 Minimal exponents m such that the fractional part of (10/9)^m obtains a maximum (when starting with m=1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 13, 17, 413, 555, 2739, 3509, 3869, 5513, 12746, 31808, 76191, 126237, 430116, 477190, 1319307, 3596185

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 (10/9)^m is greater than the fractional part of (10/9)^k for all k, 1 <= k < m.
The next such number must be greater than 2*10^5.
a(23) > 10^7. - Robert Price, Mar 24 2019

			a(7)=13, since fract((10/9)^13) = 0.93..., but fract((10/9)^k) < 0.89 for 1 <= k <= 12; thus fract((10/9)^13) > fract((10/9)^k) for 1 <= k < 13 and 13 is the minimal exponent > 6 with this property.

Cf. A153663, A153671, A153679, A153687, A153699, A154130, A091560, A153711, A153719.

$MaxExtraPrecision = 100000;
p = 0; Select[Range[1, 20000],
If[FractionalPart[(10/9)^#] > p, p = FractionalPart[(10/9)^#];
True] &] (* Robert Price, Mar 24 2019 *)

A153695_list, m, m10, m9, q = [], 1, 10, 9, 0
while m < 10**4:
    r = m10 % m9
    if r > q:
        q = r
        A153695_list.append(m)
    m += 1
    m10 *= 10
    m9 *= 9
    q *= 9 # Chai Wah Wu, May 16 2020

Recursion: a(1):=1, a(k):=min{ m>1 | fract((10/9)^m) > fract((10/9)^a(k-1))}, where fract(x) = x-floor(x).

a(19)-a(22) from Robert Price, Mar 24 2019

A153696 Numbers k such that the fractional part of (10/9)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 6, 13, 17, 5513, 12746, 126237, 430116, 477190, 1295623, 1319307, 3596185, 6109350

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract((10/9)^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 2*10^5.
a(14) > 10^7. - Robert Price, Mar 24 2019

			a(3) = 13, since fract((10/9)^13) = 0.9341... > 0.923... = 1 - (1/13), but fract((10/9)^k) <= 1 - (1/k) for 1 < k < 13.

Cf. A153664, A153672, A153680, A153688, A153700, A154130, A153704, A153712, A153720.

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

a(8)-a(13) from Hans Havermann, Apr 04 2016

A153701 Minimal exponents m such that the fractional part of e^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 2, 3, 9, 29, 45, 75, 135, 219, 732, 1351, 3315, 4795, 4920, 5469, 28414, 37373

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 e^m is less than the fractional part of e^k for all k, 1<=k

The next such number must be greater than 100000.

a(18) > 300,000. Robert Price, Mar 23 2019

			a(4)=9, since fract(e^9)=0.08392..., but fract(e^k)>=0.08553... for 1<=k<=8; thus fract(e^9)<fract(e^k) for 1<=k<9.

Cf. A153661, A153669, A153677, A153685, A153693, A153705, A154130, A137994, A153717, A000149.

$MaxExtraPrecision = 100000;
p = 1; Select[Range[1, 300000],
If[FractionalPart[E^#] < p, p = FractionalPart[E^#]; True] &] (* Robert Price, Mar 23 2019 *)

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

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

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A153685 Minimal exponents m such that the fractional part of (11/10)^m obtains a minimum (when starting with m=1).

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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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A153687 Minimal exponents m such that the fractional part of (11/10)^m obtains a maximum (when starting with m=1).

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A153688 Numbers k such that the fractional part of (11/10)^k is greater than 1-(1/k).

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A153693 Minimal exponents m such that the fractional part of (10/9)^m obtains a minimum (when starting with m=1).

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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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A153695 Minimal exponents m such that the fractional part of (10/9)^m obtains a maximum (when starting with m=1).

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Mathematica

Python

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Extensions

A153696 Numbers k such that the fractional part of (10/9)^k is greater than 1-(1/k).

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