A145204 Numbers whose representation in base 3 (A007089) ends in an odd number of zeros.

0, 3, 6, 12, 15, 21, 24, 27, 30, 33, 39, 42, 48, 51, 54, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 108, 111, 114, 120, 123, 129, 132, 135, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 189, 192, 195, 201, 204, 210, 213, 216, 219, 222, 228, 231

Offset: 1

Author

Reinhard Zumkeller, Oct 04 2008

Keywords

• nonn

Comments

Previous name: Complement of A007417.

Also numbers having infinitary divisor 3, or the same, having factor 3 in their Fermi-Dirac representation as product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013

For n > 1: where even terms occur in A051064. - Reinhard Zumkeller, May 23 2013

If we exclude a(1) = 0, these are numbers whose squarefree part is divisible by 3, which can be partitioned into numbers whose squarefree part is congruent to 3 mod 9 (A055041) and 6 mod 9 (A055040) respectively. - Peter Munn, Jul 14 2020

The inclusion of 0 as a term might be viewed as a cultural preference: if we habitually wrote numbers enclosed in brackets and then used a null string of digits for zero, the natural number sequence in ternary would be [], [1], [2], [10], [11], [12], [20], ... . - Peter Munn, Aug 02 2020

The asymptotic density of this sequence is 1/4. - Amiram Eldar, Sep 20 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
• Index entries for 3-automatic sequences.

Crossrefs

Subsequence of A008585, A028983.

Subsequences: A016051, A055040, A055041, A329575.

Cf. A007089, A007417 (complement), A050376, A182581 (characteristic function).

Positions of 0s in A014578.

Excluding 0: the positions of odd numbers in A007949; equivalently, of even numbers in A051064; symmetric difference of A003159 and A036668.

Related to A042964 via A052330.

Related to A036554 via A064614.

Programs

Haskell

a145204 n = a145204_list !! (n-1)
a145204_list = 0 : map (+ 1) (findIndices even a051064_list)
-- Reinhard Zumkeller, May 23 2013

Maple

isA145204 := proc(n) local d, c;
if n = 0 then return true fi;
d := A007089(n); c := 0;
while irem(d, 10) = 0 do c := c+1; d := iquo(d, 10) od;
type(c, odd) end:
select(isA145204, [$(0..231)]); # Peter Luschny, Aug 05 2020

Mathematica

Select[ Range[0, 235], (# // IntegerDigits[#, 3]& // Split // Last // Count[#, 0]& // OddQ)&] (* Jean-François Alcover, Mar 18 2013 *)
Join[{0}, Select[Range[235], OddQ @ IntegerExponent[#, 3] &]] (* Amiram Eldar, Sep 20 2020 *)

Python

import numpy as np
def isA145204(n):
    if n == 0: return True
    c = 0
    d = int(np.base_repr(n, base = 3))
    while d % 10 == 0:
        c += 1
        d //= 10
    return c % 2 == 1
print([n for n in range(231) if isA145204(n)]) # Peter Luschny, Aug 05 2020

Python

from sympy import integer_log
def A145204(n):
    if n == 1: return 0
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 15 2025

Formula

a(n) = 3 * A007417(n-1) for n > 1.

A014578(a(n)) = 0.

For n > 1, A007949(a(n)) mod 2 = 1. [Edited by Peter Munn, Aug 02 2020]

{a(n) : n >= 2} = {A052330(A042964(k)) : k >= 1} = {A064614(A036554(k)) : k >= 1}. - Peter Munn, Aug 31 2019 and Dec 06 2020

Extensions

New name using a comment of Vladimir Shevelev by Peter Luschny, Aug 05 2020