A094969 -id:A094969 - cp's OEIS frontend

A002379 a(n) = floor(3^n / 2^n).

1, 1, 2, 3, 5, 7, 11, 17, 25, 38, 57, 86, 129, 194, 291, 437, 656, 985, 1477, 2216, 3325, 4987, 7481, 11222, 16834, 25251, 37876, 56815, 85222, 127834, 191751, 287626, 431439, 647159, 970739, 1456109, 2184164, 3276246, 4914369, 7371554, 11057332

Offset: 0

N. J. A. Sloane

It is an important unsolved problem related to Waring's problem to show that a(n) = floor((3^n-1)/(2^n-1)) holds for all n > 1. This has been checked for 10000 terms and is true for all sufficiently large n, by a theorem of Mahler. [Lichiardopol]
a(n) = floor((3^n-1)/(2^n-1)) holds true at least for 2 <= n <= 305000. - Hieronymus Fischer, Dec 31 2008
a(n) is also the curve length (rounded down) of the Sierpiński arrowhead curve after n iterations, let a(0) = 1. - Kival Ngaokrajang, May 21 2014
a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 5, 7 or 11 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

R. K. Guy, Unsolved Problems in Number Theory, E19.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
S. S. Pillai, On Waring's problem, J. Indian Math. Soc., 2 (1936), 16-44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. Lichiardopol, Problem 925 (BCC20.19), A number-theoretic problem, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
K. Mahler, On the fractional parts of the powers of a rational number, II, Mathematika 4 (1957), 122-124.
Kival Ngaokrajang, Illustration of Sierpinski arrowhead curve for n = 0..5
Eric Weisstein's World of Mathematics, Power Floors
Wikipedia, Sierpiński arrowhead curve

Cf. A094969-A094500, A000217, A081464, A153662, A153665, A153666.
Cf. A060692, A002380, A000079, A000244.
Cf. A046037, A070758, A070759, A067904 (Composites and Primes).
Cf. A064628 (an analog for 4/3).

a002379 n = 3^n `div` 2^n  -- Reinhard Zumkeller, Jul 11 2014

[Floor(3^n / 2^n): n in [0..40]]; // Vincenzo Librandi, Sep 08 2011

A002379:=n->floor(3^n/2^n); seq(A002379(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[(3/2)^n], {n, 0, 40}] (* Robert G. Wilson v, May 11 2004 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; Array[x, 40] (* Fred Daniel Kline, Dec 21 2017 *)
x[n_]:=Round[-(1/2) + (3^n - 1)/(2^n - 1)]; Array[x, 39, 2] (* offset n+1, Fred Daniel Kline, Apr 13 2018 *)

makelist(floor(3^n/2^n), n, 0, 50); /* Martin Ettl, Oct 17 2012 */

a(n)=3^n>>n \\ Charles R Greathouse IV, Jun 10 2011

def A002379(n): return 3**n>>n # Chai Wah Wu, Sep 21 2022

a(n) = b(n) - (-2/3)^n where b(n) is defined by the recursion b(0):=2, b(1):=5/6, b(n+1):=(5/6)*b(n) + b(n-1). - Hieronymus Fischer, Dec 31 2008
a(n) = (1/2)*(b(n) + sqrt(b(n)^2 - (-4)^n)) (with b(n) as defined above). - Hieronymus Fischer, Dec 31 2008
3^n = a(n)*2^n + A002380(n). - R. J. Mathar, Oct 26 2012
a(n) = -(1/2) + (3/2)^n + arctan(cot((3/2)^n Pi)) / Pi. - Fred Daniel Kline, Apr 14 2018
a(n+1) = round( -(1/2) + (3^n-1)/(2^n-1) ). - Fred Daniel Kline, Apr 14 2018

More terms from Robert G. Wilson v, May 11 2004

A094999 a(n) = floor(12^n/11^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 21, 22, 25, 27, 29, 32, 35, 38, 42, 45, 50, 54, 59, 65, 71, 77, 84, 92, 100, 109, 119, 130, 142, 155, 169, 185, 201, 220, 240, 262, 285, 311, 340, 371, 404, 441, 481

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A002379, A094969-A094997.

[Floor(12^n / 11^n): n in [0..80]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(12/11)^n], {n, 0, 71}]
```

A094999(n):=floor(12^n/11^n)$ makelist(A094999(n),n,0,60); /* Martin Ettl, Oct 25 2012 */

a(n)=12^n\11^n \\ Charles R Greathouse IV, Feb 04 2016

It appears that a(n) = floor(2^(n/8)). - Yakov Shusterman (yakov(AT)mobli.com), Dec 07 2008

A064628 a(n) = floor((4/3)^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 17, 23, 31, 42, 56, 74, 99, 133, 177, 236, 315, 420, 560, 747, 996, 1328, 1771, 2362, 3149, 4199, 5599, 7466, 9954, 13273, 17697, 23596, 31462, 41950, 55933, 74577, 99437, 132583, 176777, 235703, 314271, 419028, 558704

Offset: 0

Labos Elemer, Oct 01 2001

a(n) is the perimeter of a hexaflake (rounded down) after n iterations. The total number of holes = A000420(n) - 1. The total number of irregular polygon holes = A000420(n-1) - 1. The total number of triangle holes = 6*A000420(n-1). - Kival Ngaokrajang, Apr 18 2014

a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 3, 5 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

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

Harry J. Smith, Table of n, a(n) for n=0,...,400
Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
Kival Ngaokrajang, Illustration of hexaflake for n = 0..3.
Eric Weisstein's World of Mathematics, Power Floors.
Wikipedia, n-flake.

Cf. A002379, A002380, A060692.
Cf. A094969 - A094500.
Cf. A046038, A070761, A070762, A067905 (Composites and Primes).

A064628:=n->floor(4^n/3^n); seq(A064628(n), n=0..30); # Wesley Ivan Hurt, Apr 19 2014

Table[Floor[(4/3)^n], {n, 0, 30}] (* Robert G. Wilson v *)

{ f=t=1; for (n=0, 400, write("b064628.txt", n, " ", f\t); f*=4; t*=3 ) } \\ Harry J. Smith, Sep 20 2009

def A064628(n): return floor((4/3)^n)
[A064628(n) for n in range(0, 46)] # Stefano Spezia, Oct 13 2024

More terms from Robert G. Wilson v, May 26 2004

OFFSET changed from 1 to 0 by Harry J. Smith, Sep 20 2009

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

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

A094970 a(n) = floor(7^n/2^n).

Original entry on oeis.org

1, 3, 12, 42, 150, 525, 1838, 6433, 22518, 78815, 275854, 965491, 3379220, 11827271, 41395451, 144884079, 507094277, 1774829971, 6211904899, 21741667147, 76095835015, 266335422555, 932173978944, 3262608926305, 11419131242070, 39966959347247, 139884357715364, 489595252003776

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002379, A094969-A094999.

[Floor(7^n / 2^n): n in [0..30]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(7/2)^n], {n, 0, 30}]
```

A094970(n):=floor(7^n/2^n)$ makelist(A094970(n),n,0,30); /* Martin Ettl, Oct 25 2012 */

a(n)=7^n>>n \\ Charles R Greathouse IV, Feb 04 2016

7^n = a(n)*2^n + A138616(n). - R. J. Mathar, Oct 26 2012

A091946 a(n) = floor(11^n/10^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 23, 25, 28, 30, 34, 37, 41, 45, 49, 54, 60, 66, 72, 80, 88, 97, 106, 117, 129, 142, 156, 171, 189, 207, 228, 251, 276, 304, 334, 368, 405, 445, 490, 539, 593, 652, 717, 789

Offset: 0

Reinhard Zumkeller, Feb 16 2004

			a(2) = floor(1.1^2) = floor(1.21) = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A001020, A011557.

Cf. A002379, A094969-A094500, A091947.

[Floor(11^n / 10^n): n in [0..70]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(11/10)^n], {n, 0, 70}]
```

a(n) = floor(1.1^n) = floor(A001020(n)/A011557(n)).

More terms from Robert G. Wilson v, May 26 2004

A094976 a(n) = floor(8^n/3^n).

Original entry on oeis.org

1, 2, 7, 18, 50, 134, 359, 958, 2557, 6818, 18183, 48490, 129307, 344820, 919522, 2452059, 6538825, 17436866, 46498311, 123995496, 330654658, 881745755, 2351322014, 6270192038, 16720512102, 44588032273, 118901419396

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002379, A094969-A094999.

[Floor(8^n / 3^n): n in [0..40]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(8/3)^n], {n, 0, 30}]
```

a(n) = 8^n\3^n; \\ Michel Marcus, Oct 05 2017

A094997 a(n) = floor(11^n/9^n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 9, 11, 13, 16, 20, 24, 30, 37, 45, 55, 67, 82, 101, 123, 150, 184, 225, 275, 336, 411, 503, 614, 751, 918, 1122, 1372, 1677, 2049, 2505, 3062, 3742, 4574, 5590, 6832, 8351, 10207, 12475, 15247, 18636, 22777, 27839, 34025, 41587

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A002379, A094969-A094999.

[Floor(11^n / 9^n): n in [0..70]]; // Vincenzo Librandi, Sep 09 2011

Mathematica
```
Table[ Floor[(11/9)^n], {n, 0, 30}]
```

A094971 a(n) = floor(9^n/2^n).

Original entry on oeis.org

1, 4, 20, 91, 410, 1845, 8303, 37366, 168151, 756680, 3405062, 15322783, 68952523, 310286355, 1396288601, 6283298708, 28274844190, 127236798856, 572565594852, 2576545176835, 11594453295761, 52175039830928, 234787679239180

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A002379, A094969-A094999.

[Floor(9^n / 2^n): n in [0..30]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(9/2)^n], {n, 0, 30}]
```

a(n)=9^n>>n \\ Charles R Greathouse IV, Feb 04 2016

cp's OEIS Frontend

A002379 a(n) = floor(3^n / 2^n).

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A094999 a(n) = floor(12^n/11^n).

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A064628 a(n) = floor((4/3)^n).

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

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

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A094970 a(n) = floor(7^n/2^n).

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Mathematica

Maxima

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