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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

• 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).

Links

• 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

Crossrefs

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).

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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 *)

Maxima

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

PARI

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

Python

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

Formula

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

Extensions

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