A002379 -id:A002379 - cp's OEIS frontend

A153668 Greatest number m such that the fractional part of (3/2)^A153664(n) >= 1-(1/m).

2, 14, 222, 1772, 2747, 12347, 135794, 90529, 222246, 570361, 2134829, 6901329, 4600886, 3067257, 5380892, 75503109, 814558605, 543039070, 362026046, 241350697, 160900465, 107266976, 101721580, 190708740, 127139160

Offset: 1

Hieronymus Fischer, Dec 31 2008

nonn

			a(2)=14, since 1-(1/15)=0.933...>fract((3/2)^A153664(2))=fract((3/2)^14)=0.929...>=1-(1/14).

Cf. A002379, A081464, A153662, A153663, A153664, A153665, A153666, A153667.

A153664 = {1, 14, 163, 1256, 2677, 8093, 49304, 49305, 158643, 164000, 835999, 2242294, 2242295, 2242296, 3965133, 25380333, 92600006, 92600007, 92600008, 92600009, 92600010, 92600011, 9267816, 125040717, 125040718};
Table[fp = FractionalPart[(3/2)^A153664[[n]]]; m = Floor[1/(1 - fp)];
While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153664]}] (* Robert Price, Mar 26 2019 *)

a(n) = floor(1/(1-fract((3/2)^A153664(n)))), where fract(x) = x-floor(x).

a(11)-a(25) from Robert Price, May 10 2012

A034082 a(n) = least integer m such that the part after the decimal point of the n-th root of m starts with the digit 5.

Original entry on oeis.org

21, 4, 6, 8, 12, 18, 26, 39, 58, 87, 130, 195, 292, 438, 657, 986, 1478, 2217, 3326, 4988, 7482, 11223, 16835, 25252, 37877, 56816, 85223, 127835, 191752, 287627, 431440, 647160, 970740, 1456110, 2184165, 3276247, 4914370, 7371555, 11057333

Offset: 2

Patrick De Geest, Sep 15 1998

			a(25)=25252 -> 25252^(1/25)=1.{5}000019762083...

Cf. A034062, A034072, A002379, A061418, A061419.

def A034082(n): return (3**n>>n)+1 if n > 2 else 21 # Chai Wah Wu, Sep 21 2022

For n > 2: a(n) = ceiling((3/2)^n) = A002379(n) + 1. - Henry Bottomley, May 02 2001

Definition clarified by N. J. A. Sloane, May 24 2021

A094500 Least number k such that (n+1)^k / n^k >= 2.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47, 48, 49, 49, 50, 51, 51

Offset: 1

Robert G. Wilson v, May 26 2004

This sequence also describes the minimum number of (n+1)-player games, where each player has an equal chance of winning, that must be played for a given player to have at least a 50% chance of winning at least once. E.g., a(3) = 3 because in a 4-player random game, a given player will have a greater than 50% chance of winning at least once if 3 games are played. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 28 2006
Also, a(n) denotes a median m of the geometric random variable on the positive integers with mean value n+1. The median is obtained by solving 1-(n/n+1)^m >= 1/2 for least integer m. - Dennis P. Walsh, Aug 13 2012
The limit n -> inf. a(n)/n = log 2. - Robert G. Wilson v, May 13 2014

			a(3) = 3 because (4/3)^2 < 2 and (4/3)^3 > 2.

Jon Eivind Vatne, The sequence of middle divisors is unbounded, Journal of Number Theory, Volume 172, March 2017, Pages 413-415. See n(i) p. 414.

Cf. A002379, A094969-A094999.

f[n_] := Block[{k = 1}, While[((n + 1)/n)^k < 2, k++]; k]; Array[f, 75]
(* to view the limit *) Array[ f/# &, 1000] (* Robert G. Wilson v, May 13 2014 *)

a(n)=ceil(log(2)/log(1+1/n)) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = n*log(2) + O(1). - Charles R Greathouse IV, Sep 02 2015

Edited by Jon E. Schoenfield, Apr 26 2014

A070759 n for which floor((3/2)^n) is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 21, 22, 98, 106, 164, 189, 219, 364, 443, 670, 775, 1919, 2204, 2715, 3692, 4228, 4912, 10466, 12300, 14087, 21659, 28170, 55832, 66577, 87309, 87505, 98144, 167512, 259517

Offset: 1

Eric W. Weisstein, May 04 2002

nonn

No more terms through 500000. - Ryan Propper, Dec 28 2008

R. K. Guy, Unsolved Problems in Number Theory, E19

Eric Weisstein's World of Mathematics, Power Floors

Cf. A070758, A067904, A070762.

Indices of primes in A002379.

Do[ If[ PrimeQ[ Floor[(3/2)^n]], Print[n]], {n, 1, 12500}]

One more term from Ralf Stephan, Oct 13 2002
Corrected and extended by Robert G. Wilson v, Jan 15 2003
More terms from Ryan Propper, Jan 25 2008
6 more terms from Ryan Propper, Dec 28 2008

A264918 Decimal expansion of constant z = Sum_{n>=1} {(3/2)^n} / 2^n, where {x} denotes the fractional part of x.

Original entry on oeis.org

3, 9, 3, 1, 8, 8, 4, 7, 7, 0, 4, 9, 6, 4, 4, 3, 2, 4, 4, 9, 7, 2, 5, 8, 2, 1, 3, 1, 3, 8, 9, 0, 3, 8, 8, 5, 8, 5, 4, 8, 3, 9, 1, 4, 0, 7, 8, 8, 6, 6, 2, 8, 6, 9, 5, 3, 9, 2, 9, 3, 2, 4, 7, 5, 7, 5, 7, 8, 7, 7, 5, 8, 3, 3, 8, 9, 7, 4, 9, 8, 6, 6, 8, 1, 0, 9, 7, 6, 6, 6, 2, 0, 6, 1, 0, 1, 8, 5, 8, 8, 8, 0, 1, 3, 3, 3, 0, 0, 8, 0, 5, 9, 3, 2, 2, 6, 3, 1, 5, 3, 2, 6, 8, 0, 9, 0, 4, 7, 5, 0, 4, 9, 4, 2, 6, 6, 6, 1, 2, 1, 1, 4, 2, 4, 3, 3, 4, 9, 8, 4, 4, 3, 5, 8, 4, 7, 7, 5, 8, 5, 0, 6, 5, 5, 9, 3, 3, 7, 2, 5, 0, 9, 1, 4, 3, 2, 8, 8, 7, 7, 0, 5, 4, 3, 2, 2, 3, 1, 4, 0, 7, 7, 1, 7, 1, 7, 5, 9, 5, 3, 3, 3, 7, 7, 6

Offset: 1

Paul D. Hanna, Dec 03 2015

			z = 0.39318847704964432449725821313890388585483914078866\
28695392932475757877583389749866810976662061018588\
80133300805932263153268090475049426661211424334984\
43584775850655933725091432887705432231407717175953\
33776901692614854937460993931094741172922114373160\
19617637538747813543456758934332723336245738884968...
INFINITE SERIES.
(1) z = 1/4 + 1/4^2 + 3/4^3 + 1/4^4 + 19/4^5 + 25/4^6 + 11/4^8 + 161/4^9 + 227/4^10 + 681/4^11 + 1019/4^12 +...+ A002380(n)/4^n +...
(2) 3 - z = 1/2 + 2/2^2 + 3/2^3 + 5/2^4 + 7/2^5 + 11/2^6 + 17/2^7 + 25/2^8 + 38/2^9 + 57/2^10 + 86/2^11 + 129/2^12 + 194/2^13 + 291/2^14 +...+ A002379(n)/2^n +...
where
3 - z = 2.60681152295035567550274178686109611414516...

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A002379 ([(3/2)^n]), A002380 (3^n mod 2^n), A264919, A264920, A264921, A264922.

z = Sum_{n>=1} (3^n mod 2^n) / 4^n = Sum_{n>=1} A002380(n) / 4^n.

3 - z = Sum_{n>=1} [(3/2)^n] / 2^n = Sum_{n>=1} A002379(n) / 2^n, where [x] denotes the integer floor function of x.

A065565 a(n) = floor((5/4)^n).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 5, 7, 9, 11, 14, 18, 22, 28, 35, 44, 55, 69, 86, 108, 135, 169, 211, 264, 330, 413, 516, 646, 807, 1009, 1262, 1577, 1972, 2465, 3081, 3851, 4814, 6018, 7523, 9403, 11754, 14693, 18367, 22958, 28698, 35873, 44841, 56051, 70064, 87581

Offset: 0

Benoit Cloitre, Nov 30 2001

nonn

a(n) is also the curvature (rounded down) of the circle inscribed in the n-th 3:4:5 triangle arranged in a spiral as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 21 2013

By the result of Dubickas and Novikas, a(n) is divisible by at least one of 2, 3, 7, 11, 13 infinitely often, so that a(n) is composite infinitely often. - Tomohiro Yamada, Apr 23 2017

Harry J. Smith, Table of n, a(n) for n = 0..400
Arturas Dubickas, Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635-648, available from the first author's page.
Kival Ngaokrajang, Illustration of some initial terms.

Cf. A002379, A094969-A094500.

Cf. A064628. - Tomohiro Yamada, Apr 23 2017

Table[ Floor[(5/4)^n], {n, 0, 41}] (* Robert G. Wilson v, May 26 2004 *)

a(n) = floor((5/4)^n) \\ Harry J. Smith, Oct 22 2009

Edited by N. J. A. Sloane at the suggestion of Stefan Steinerberger, Jun 20 2007

Offset changed from 1 to 0 by Harry J. Smith, Oct 22 2009

A070758 Values of floor((3/2)^n) that are composite.

Original entry on oeis.org

25, 38, 57, 86, 129, 194, 291, 437, 656, 985, 1477, 2216, 3325, 11222, 16834, 25251, 37876, 56815, 85222, 127834, 191751, 287626, 431439, 647159, 970739, 1456109, 2184164, 3276246, 4914369, 7371554, 11057332, 16585998, 24878997, 37318496

Offset: 1

Eric W. Weisstein, May 04 2002

nonn

R. K. Guy, Unsolved Problems in Number Theory, E19

Robert Israel, Table of n, a(n) for n = 1..5649
Eric Weisstein's World of Mathematics, Power Floors

Cf. A070759, A067904.

Composites in A002379.

remove(isprime, [seq(floor((3/2)^n),n=2..100)]); # Robert Israel, Oct 30 2019

Select[Floor[(3/2)^Range[50]],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 11 2017 *)

A071532 a(n) = (-1) * Sum_{k=1..n} (-1)^floor((3/2)^k).

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 9, 8, 7, 8, 7, 8, 9, 10, 11, 10, 9, 10, 9, 8, 7, 8, 7, 6, 7, 8, 7, 8, 7, 6, 5, 6, 7, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 12, 11, 10, 11, 10, 11, 10, 9, 10, 9, 10, 9, 8, 7, 6, 5, 6, 7, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5, 4, 3

Offset: 1

Benoit Cloitre, Jun 20 2002

Let b(n) denote the number of k with 0<=k<=n such that floor((3/2)^k) = A002379(k) is even; then a(n) = n-2*b(n).
Equivalently: let c(n) denote the number of k, 0<=k<=n, such that floor((3/2)^k) = A002379(k) is odd, then a(n) = 2*c(n)-n.
Is a(n)>0? For n large enough does a(n)>sqrt(n) always hold?
Conjecture: asymptotically, a(n) ~ C * Log(n)^2 with C = 1.4.....

Robert G. Wilson v, Graph of first 100000 terms

Cf. A002379, A072418.

a[0] = 0; a[n_] := a[n] = a[n - 1] - (-1)^Floor[(3/2)^n]; Table[ a[n], {n, 0, 95}]

a(n)=-sum(i=1, n, sign((-1)^floor((3/2)^i)))

a(n)=n-2*sum(k=0,n,if(floor((3/2)^k)%2,0,1))

a(n) = (-1) * Sum_{i=1..n} (-1)^A002379(i).

Edited by Ralf Stephan, Sep 01 2004

A082511 a(n) = 3^n mod 2n.

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 3, 1, 9, 9, 3, 9, 3, 9, 27, 1, 3, 9, 3, 1, 27, 9, 3, 33, 43, 9, 27, 25, 3, 9, 3, 1, 27, 9, 47, 9, 3, 9, 27, 1, 3, 57, 3, 81, 63, 9, 3, 33, 31, 49, 27, 81, 3, 81, 67, 65, 27, 9, 3, 81, 3, 9, 27, 1, 113, 69, 3, 81, 27, 109, 3, 81, 3, 9, 57, 81, 75, 105, 3, 1, 81, 9, 3, 57, 73

Offset: 1

Labos Elemer, Apr 28 2003

nonn

			Residues are often also powers of 3, that is, 3^n = k*2*n + 3^j, as is the case for n=1..23. The first terms that are not powers of 3 are a(24)=33 and a(25)=43.
a(6)=9: modulus = 2*n = 12; 3^n = 3^6 = 729 = 60*12 + 9 = 720 + a(6).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A000079, A000244, A002379.

Cf. A083528, A083529, A083530.

Table[PowerMod[3,n,2n],{n,90}] (* Harvey P. Dale, Jan 21 2014 *)

a(n) = lift(Mod(3, 2*n)^n) \\ Felix Fröhlich, Oct 20 2018

for n in range(1, 80): print(pow(3, n, 2*n), end=" ") # Stefano Spezia, Oct 20 2018

A170827 Sum of digits after the decimal point in the decimal expansion of (3/2)^n.

Original entry on oeis.org

0, 5, 7, 15, 13, 29, 25, 37, 38, 43, 42, 67, 60, 67, 60, 85, 73, 77, 71, 79, 95, 107, 106, 100, 95, 120, 95, 137, 143, 146, 138, 140, 147, 166, 172, 163, 172, 177, 180, 193, 158, 174, 171, 184, 177, 188, 188, 223, 212, 241, 213, 198, 243, 229, 236, 245, 278, 281, 305, 304

Offset: 0

Michel Lagneau, Dec 28 2009

Some terms appear more than once, 60 being the first. - Robert G. Wilson v, Mar 10 2015

			1.0
1.5
2.25
3.375
5.0625
7.59375
11.390625
17.0859375
25.62890625

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A002379, A170828.

td[n_]:=Module[{rd=RealDigits[(3/2)^n]},Total[Drop[rd[[1]], rd[[2]]]]]; Array[td,60,0] (* Harvey P. Dale, Dec 13 2011 *)

a(n) = sumdigits(lift(Mod(15,10^n)^n)) \\ Jianing Song, Sep 28 2022

Edited by N. J. A. Sloane, Dec 28 2009

cp's OEIS Frontend

A153668 Greatest number m such that the fractional part of (3/2)^A153664(n) >= 1-(1/m).

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A034082 a(n) = least integer m such that the part after the decimal point of the n-th root of m starts with the digit 5.

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A094500 Least number k such that (n+1)^k / n^k >= 2.

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A070759 n for which floor((3/2)^n) is prime.

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A264918 Decimal expansion of constant z = Sum_{n>=1} {(3/2)^n} / 2^n, where {x} denotes the fractional part of x.

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A065565 a(n) = floor((5/4)^n).

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A070758 Values of floor((3/2)^n) that are composite.

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A071532 a(n) = (-1) * Sum_{k=1..n} (-1)^floor((3/2)^k).

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