A070759 -id:A070759 - 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

A070762 n for which floor((4/3)^n) is prime.

Original entry on oeis.org

3, 4, 6, 7, 9, 10, 11, 12, 38, 42, 59, 96, 154, 171, 211, 313, 465, 563, 1040, 1176, 1213, 1431, 1519, 1987, 2527, 3033, 4039, 4209, 4358, 5109, 5251, 6642, 19200, 25275, 42589, 43025, 49294, 58585, 66290, 77458, 80409, 86533, 94192, 110452, 115166, 124470

Offset: 1

Eric W. Weisstein, May 04 2002

nonn

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

Giovanni Resta, Table of n, a(n) for n = 1..53 (terms < 400000)
Eric Weisstein's World of Mathematics, Power Floors

Cf. A046038, A067905, A070761, A070759.

Do[ If[ PrimeQ[ Floor[(4/3)^n]], Print[n]], {n, 1, 17500}]
Select[Range[7000],PrimeQ[Floor[(4/3)^#]]&] (* The program generates the first 32 terms of the sequence. *) (* Harvey P. Dale, Oct 02 2024 *)

Corrected by Robert G. Wilson v, Jan 15 2003

More terms from Ryan Propper, Jan 25 2008

A067904 Primes of the form floor((3/2)^k).

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 4987, 7481, 180693856682317883, 4630985912862061063, 75677449184722757264165738713, 1910944005427272291238064043761449, 366425537175409658704814112327931286021

Offset: 1

Benoit Cloitre, Mar 03 2002

nonn

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..23

Cf. A043037, A070758, A070759.

f[n_]:=Floor[(3/2)^n]; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,p]],{n,0,4*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 02 2009 *)
Select[Floor[(3/2)^Range[300]],PrimeQ] (* Harvey P. Dale, Dec 28 2014 *)

v=[];for(k=2,5700, if(ispseudoprime(t=floor((3/2)^k)), v=concat(v,t))); v \\ Charles R Greathouse IV, Feb 15 2011

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 *)

A072924 Least k such that floor((1+1/k)^n) is prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 3, 6, 8, 7, 6, 6, 7, 5, 11, 2, 2, 9, 4, 6, 10, 5, 9, 5, 6, 4, 7, 10, 11, 7, 6, 4, 3, 10, 4, 4, 3, 5, 4, 17, 6, 11, 7, 5, 14, 12, 8, 6, 11, 4, 14, 8, 7, 3, 16, 4, 21, 8, 12, 7, 8, 7, 7, 18, 12, 8, 17, 10, 12, 28, 6, 24, 16, 12, 16, 18, 7, 6, 6, 7, 11, 8, 14, 24, 8

Offset: 1

Benoit Cloitre, Aug 11 2002

a(n) = 2 for n in A070759. a(n) = 3 for n in A070762 but not in A070759. - Robert Israel, Jan 09 2018

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

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Scatter plot of a(n)/sqrt(n) for n=1..15000

Cf. A070759, A070762

f:= proc(n) local k;
  for k from 1 do if isprime(floor((1+1/k)^n)) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 09 2018

lkp[n_]:=Module[{k=1},While[!PrimeQ[Floor[(1+1/k)^n]],k++];k]; Array[ lkp,90] (* Harvey P. Dale, Dec 02 2018 *)

a(n)=if(n<0,0,s=1; while(isprime(floor((1+1/s)^n)) == 0,s++); s)

It seems that a(n)/sqrt(n) is bounded. More precisely for n large enough it seems that (1/2)*sqrt(n) < a(n) < 3*sqrt(n).

On the contrary, A.L. Whiteman conjectured that the sequence floor(r^n) for non-integer rational r > 1 always contains infinitely many primes. If this conjecture is true for some r=1+1/k, then lim inf_{n -> infinity} a(n) is finite. - Robert Israel, Jan 09 2018

cp's OEIS Frontend

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

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A070762 n for which floor((4/3)^n) is prime.

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A067904 Primes of the form floor((3/2)^k).

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

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Mathematica

A072924 Least k such that floor((1+1/k)^n) is prime.

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Programs

Maple

Mathematica

PARI

Formula