A032924 Numbers whose ternary expansion contains no 0.

1, 2, 4, 5, 7, 8, 13, 14, 16, 17, 22, 23, 25, 26, 40, 41, 43, 44, 49, 50, 52, 53, 67, 68, 70, 71, 76, 77, 79, 80, 121, 122, 124, 125, 130, 131, 133, 134, 148, 149, 151, 152, 157, 158, 160, 161, 202, 203, 205, 206, 211, 212, 214, 215, 229, 230, 232, 233, 238, 239

Offset

1,2

Comments

Complement of A081605. - Reinhard Zumkeller, Mar 23 2003

Subsequence of A154314. - Reinhard Zumkeller, Jan 07 2009

The first 28 terms are the range of A059852 (Morse codes for letters, when written in base 3) union {44, 50} (which correspond to Morse codes of Ü and Ä). Subsequent terms represent the Morse code of other symbols in the same coding. - M. F. Hasler, Jun 22 2020

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007), Article 07.1.5., 1-13.

Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.

Index entries for 3-automatic sequences.

Formula

a(n) = A107680(n) + A107681(n). - Reinhard Zumkeller, May 20 2005

A081604(A107681(n)) <= A081604(A107680(n)) = A081604(a(n)) = A000523(n+1). - Reinhard Zumkeller, May 20 2005

A077267(a(n)) = 0. - Reinhard Zumkeller, Mar 02 2008

a(1)=1, a(n+1) = f(a(n)+1,f(a(n)+1) where f(x,y) = if x<3 and x<>0 then y, else if x mod 3 = 0 then f(y+1,y+1), else f(floor(x/3),y). - Reinhard Zumkeller, Mar 02 2008

a(2*n) = a(2*n-1)+1, n>0. - Zak Seidov, Jul 27 2009

A212193(a(n)) = 0. - Reinhard Zumkeller, May 04 2012

a(2*n+1) = 3*a(n)+1. - Robert Israel, Aug 05 2015

G.f.: x/(1-x)^2 + Sum_{m >= 1} 3^(m-1)*x^(2^(m+1)-1)/((1-x^(2^m))*(1-x))). - Robert Israel, Aug 04 2015

A065361(a(n)) = n. - Rémy Sigrist, Feb 06 2023

Maple

f:= proc(n) local L, i, m;
   L:= convert(n, base, 2);
   m:= nops(L);
   add((1+L[i])*3^(i-1), i=1..m-1);
end proc:
map(f, [$2..101]); # Robert Israel, Aug 04 2015

Mathematica

Select[Range@ 240, Last@ DigitCount[#, 3] == 0 &] (* Michael De Vlieger, Aug 05 2015 *)
Flatten[Table[FromDigits[#, 3]&/@Tuples[{1, 2}, n], {n, 5}]] (* Harvey P. Dale, May 28 2016 *)

Prog

(Haskell)
a032924 n = a032924_list !! (n-1)
a032924_list = iterate f 1 where
   f x = 1 + if r < 2 then x else 3 * f x'  where (x', r) = divMod x 3
-- Reinhard Zumkeller, Mar 07 2015, May 04 2012
(PARI) apply( {A032924(n)=if(n<3, n, 3*self()((n-1)\2)+2-n%2)}, [1..99]) \\ M. F. Hasler, Jun 22 2020
(PARI) a(n) = fromdigits(apply(d->d+1, binary(n+1)[^1]), 3); \\ Kevin Ryde, Jun 23 2020
(Python)
def a(n): return sum(3**i*(int(b)+1) for i, b in enumerate(bin(n+1)[:2:-1]))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Aug 15 2022
(Python)
def is_A032924(n):
    while n > 2:
       n, r = divmod(n, 3)
       if r==0: return False
    return n > 0
print([n for n in range(250) if is_A032924(n)]) # M. F. Hasler, Feb 15 2023
(Python)
def A032924(n): return int(bin(m:=n+1)[3:], 3) + (3**(m.bit_length()-1)-1>>1) # Chai Wah Wu, Oct 13 2023

Crossrefs

Cf. A005823, A005836, A065361, A007089, A081608, A132140, A132141.

Zeroless numbers in some other bases <= 10: A000042 (base 2), A023705 (base 4), A248910 (base 6), A255805 (base 8), A255808 (base 9), A052382 (base 10).

Sequence in context: A072920 A330708 A107899 * A005125 A232832 A272195

Adjacent sequences: A032921 A032922 A032923 * A032925 A032926 A032927

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved