A212193 -id:A212193 - cp's OEIS frontend

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

Clark Kimberling

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
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.

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).

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

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

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 *)

apply( {A032924(n)=if(n<3,n,3*self()((n-1)\2)+2-n%2)}, [1..99]) \\ M. F. Hasler, Jun 22 2020

a(n) = fromdigits(apply(d->d+1,binary(n+1)[^1]), 3); \\ Kevin Ryde, Jun 23 2020

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

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

def A032924(n): return int(bin(m:=n+1)[3:],3) + (3**(m.bit_length()-1)-1>>1) # Chai Wah Wu, Oct 13 2023

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,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
Sum_{n>=1} 1/a(n) = 3.4977362637842652509313189236131190039368413460747606236619907531632476445332666030262441154353753276457... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A081605 Numbers having at least one 0 in their ternary representation.

Original entry on oeis.org

0, 3, 6, 9, 10, 11, 12, 15, 18, 19, 20, 21, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 45, 46, 47, 48, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 72, 73, 74, 75, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A032924.

A212193(a(n)) <> 0. [Reinhard Zumkeller, May 04 2012]

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

Cf. A007089, A077267, A081607, A081606, A074940.

import Data.List (findIndices)
a081605 n = a081605_list !! (n-1)
a081605_list = findIndices (/= 0) a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[0,100],DigitCount[#,3,0]>0&] (* Harvey P. Dale, Aug 10 2021 *)

from itertools import count, islice
def A081605_gen(): # generator of terms
    a = -1
    for n in count(2):
        b = int(bin(n)[3:],3) + (3**(n.bit_length()-1)-1>>1)
        yield from range(a+1,b)
        a = b
A081605_list = list(islice(A081605_gen(),30)) # Chai Wah Wu, Oct 13 2023

A154314 Numbers with not more than two distinct digits in ternary representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 36, 37, 39, 40, 41, 43, 44, 49, 50, 52, 53, 54, 56, 60, 62, 67, 68, 70, 71, 72, 74, 76, 77, 78, 79, 80, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 122

Offset: 1

Reinhard Zumkeller, Jan 07 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.

Complement of A031944.
Union of A032924, A005823 and A005836.
Cf. A007089, A043530, A212193.

import Data.List (findIndices)
a154314 n = a154314_list !! (n-1)
a154314_list = findIndices (/= 3) a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[0,200],Length[Union[IntegerDigits[#,3]]]<3&] (* Harvey P. Dale, Nov 23 2012 *)

is(n)=#Set(digits(n,3))<3 \\ Charles R Greathouse IV, Mar 17 2014

A043530(a(n)) <= 2.
A212193(a(n)) <> 3. - Reinhard Zumkeller, May 04 2012
a(n) >> n^1.58..., where the exponent is log(3)/log(2). - Charles R Greathouse IV, Mar 17 2014
Sum_{n>=2} 1/a(n) = 5.47555542241781419692840472181029603722178623821762258873485212626135391726959422416350447132335696748507... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A031944 Numbers in which digits 0,1,2 all occur in base 3.

Original entry on oeis.org

11, 15, 19, 21, 29, 32, 33, 34, 35, 38, 42, 45, 46, 47, 48, 51, 55, 57, 58, 59, 61, 63, 64, 65, 66, 69, 73, 75, 83, 86, 87, 88, 89, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 110, 113, 114, 115, 116, 119, 123, 126, 127, 128, 129, 132, 135

Offset: 1

Clark Kimberling

A043530(a(n)) = 3; complement of A154314. - Reinhard Zumkeller, Jan 07 2009

A212193(a(n)) = 3. - Reinhard Zumkeller, May 04 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Pandigital Number
Wikipedia, Ternary numeral system
Wikipedia, Pandigital number

Cf. A043530, A049354, A154314, A212193.

import Data.List (elemIndices)
a031944 n = a031944_list !! (n-1)
a031944_list = elemIndices 3 a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[200],Min[DigitCount[#,3]]>0&] (* Harvey P. Dale, Nov 21 2015 *)

from sympy.ntheory import count_digits
def ok(n): c = count_digits(n, 3); return all(c[d] > 0 for d in [0, 1, 2])
print([k for k in range(136) if ok(k)]) # Michael S. Branicky, Nov 15 2021

A067898 Least digit not used in n (or 10 if n is pandigital).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 2, 2

Offset: 0

Rick L. Shepherd, May 13 2003

a(A050278(1)) = a(1023456789) = 10, the first term with that value, as 1023456789 is the first base 10 pandigital number.

a(A052382(n)) = 0; a(A011540(n)) > 0. [Reinhard Zumkeller, May 04 2012]

			a(10)=2 because decimal digits 0 and 1 are both used in 10, a(102)=3 because decimal digits 0, 1 and 2 are used in 102.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A050278 (pandigital numbers).

Cf. A212193 (ternary).

import Data.List (delete)
a067898 n = f n [0..10] where
   f x ys | x <= 9    = head $ delete x ys
          | otherwise = f x' $ delete d ys where (x',d) = divMod x 10
-- Reinhard Zumkeller, May 04 2012

def A067898(n):
    s = set(str(n))
    for i in range(10):
        if str(i) not in s:
            return i
    return 10 # Chai Wah Wu, Apr 13 2024

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A032924 Numbers whose ternary expansion contains no 0.

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A081605 Numbers having at least one 0 in their ternary representation.

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A154314 Numbers with not more than two distinct digits in ternary representation.

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A031944 Numbers in which digits 0,1,2 all occur in base 3.

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Python

A067898 Least digit not used in n (or 10 if n is pandigital).

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