A007417 If k appears, 3k does not.

1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 97, 98, 99, 100

Offset: 1

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• easy
• nonn

Comments

The characteristic function of this sequence is given by A014578. - Philippe Deléham, Mar 21 2004

Numbers whose ternary representation ends in even number of zeros. - Philippe Deléham, Mar 25 2004

Numbers for which 3 is not an infinitary divisor. - Vladimir Shevelev, Mar 18 2013

Where odd terms occur in A051064. - Reinhard Zumkeller, May 23 2013

Examples

			From _Gary W. Adamson_, Mar 02 2010: (Start)
Given the following multiplication table: top row = "not multiples of 3", left column = powers of 3; we get:
   1   2   4   5   7   8   10   11   13
   3   6  12  15  21  24   30   33   39
   9  18  36  45  63  72   90   99  114
  27  54 108
  81
If rows are labeled (1, 2, 3, ...) then odd-indexed rows are in the set; but evens not. Examples: 9 is in the set since 3 is not, but 27 in row 4 can't be. (End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
• Simon Plouffe, Email to N. J. A. Sloane, Jun. 1994
• David Wakeham and David R. Wood, On multiplicative Sidon sets, INTEGERS, 13 (2013), #A26.
• Index entries for 3-automatic sequences.

Crossrefs

Complement of A145204. - Reinhard Zumkeller, Oct 04 2008

Cf. A007949, A014578 (characteristic function), A042948, A051064, A052330, A092400, A092401.

Programs

Haskell

import Data.List (delete)
a007417 n = a007417_list !! (n-1)
a007417_list = s [1..] where
   s (x:xs) = x : s (delete (3*x) xs)

Mathematica

Select[ Range[100], (# // IntegerDigits[#, 3]& // Split // Last // Count[#, 0]& // EvenQ)&] (* Jean-François Alcover, Mar 01 2013, after Philippe Deléham *)
Select[Range[100], EvenQ@ IntegerExponent[#, 3] &] (* Michael De Vlieger, Sep 01 2020 *)

PARI

is(n) = { my(i = 0); while(n%3==0, n/=3; i++); i%2==0; } \\ Iain Fox, Nov 17 2017

PARI

is(n)=valuation(n,3)%2==0; \\ Joerg Arndt, Aug 08 2020

Python

from sympy import integer_log
def A007417(n):
    def f(x): return n+x-sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 15 2025

Formula

Limit_{n->infinity} a(n)/n = 4/3. - Philippe Deléham, Mar 21 2004

Partial sums of A092400. Indices of even numbers in A007949. Indices of odd numbers in A051064. a(n) = A092401(2n-1). - Philippe Deléham, Mar 29 2004

{a(n)} = A052330({A042948(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Aug 31 2019

Extensions

More terms from Philippe Deléham, Mar 29 2004

Typo corrected by Philippe Deléham, Apr 15 2010