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A145204 Numbers whose representation in base 3 (A007089) ends in an odd number of zeros. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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)} \ {0} = A052330({A042964(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Aug 31 2019

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-Fran├žois Alcover, Mar 18 2013 *)

Join[{0}, Select[Range[235], OddQ @ IntegerExponent[#, 3] &]] (* Amiram Eldar, Sep 20 2020 *)

PROG

(Haskell)

a145204 n = a145204_list !! (n-1)

a145204_list = 0 : map (+ 1) (findIndices even a051064_list)

-- Reinhard Zumkeller, May 23 2013

# 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

CROSSREFS

Subsequence of A008585.

Subsequences: A016051, A055040, A055041, A329575.

Cf. A007089, A007417 (complement), A007949, A014578, A042964, A050376, A051064, A052330, A182581 (characteristic function).

Sequence in context: A319451 A256882 A191267 * A016051 A070790 A114614

Adjacent sequences:  A145201 A145202 A145203 * A145205 A145206 A145207

KEYWORD

nonn,changed

AUTHOR

Reinhard Zumkeller, Oct 04 2008

EXTENSIONS

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

STATUS

approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)