A064235 The smallest power of 3 that is greater than or equal to n.

1, 3, 3, 9, 9, 9, 9, 9, 9, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81

Offset: 1

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Sep 22 2001

Keywords

• nonn

Comments

Let A_n be the upper triangular matrix in the group GL(n,3) of invertible n X n matrices over GF(3) that has zero entries below the diagonal and 1 elsewhere. For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /. a(n) is the order of this matrix as an element of GL(n,3).

For n>1 a(n) is the smallest integer such that gcd(a(n),2^a(n)+1) >= n. - Benoit Cloitre, Apr 21 2002

From Jianing Song, Jul 05 2025: (Start)

a(n+1) is the period of {binomial(N,n) mod 3: N in Z}. For the general result, see A349593.

Since the modulus (3) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem. (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Wikipedia, Lucas's theorem

Crossrefs

Cf. A062383.

Cf. A132171, A080342, A000244.

With offset 0, column 3 of A349593. A062383, A385552, A385553, and A385554 are respectively columns 2, 5, 6, and 10.

Programs

Haskell

import Data.List (transpose)
a064235 n = genericIndex a064235_list (n - 1)
a064235_list = 1 : zs where
   zs = 3 : 3 : (map (* 3) $ concat $ transpose [zs, zs, zs])
-- Reinhard Zumkeller, Sep 02 2015

Maple

A064235 := proc(n)
        ceil(log(n)/log(3)) ;
        3^% ;
end proc: # R. J. Mathar, Nov 06 2011

Mathematica

nn=100;With[{p3=3^Range[0,Ceiling[Log[3,nn]]]},Flatten[Table[Select[ p3, #>=n&, 1],{n,nn}]]] (* Harvey P. Dale, Mar 14 2013 *)

Python

from gmpy2 import digits
def A064235(n): return 3**len(digits(n-1,3)) if n>1 else 1 # Chai Wah Wu, Oct 21 2024

Formula

a(n) = 3 ^ A080342(n). - Reinhard Zumkeller, Sep 02 2015

Sum_{n>=1} 1/a(n)^2 = 4/3. - Amiram Eldar, Aug 16 2022

Extensions

More terms from James Sellers, Sep 26 2001