A154314 -id:A154314 - 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

A125292 Numbers having either no ones or no twos in their ternary representation.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 18, 20, 24, 26, 27, 28, 30, 31, 36, 37, 39, 40, 54, 56, 60, 62, 72, 74, 78, 80, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 162, 164, 168, 170, 180, 182, 186, 188, 216, 218, 222, 224, 234, 236, 240, 242, 243

Offset: 1

Reinhard Zumkeller, Nov 26 2006

Complement of A125293; union of A005823 and A005836;

A125291(a(n)) = 1; A062756(a(n))*A081603(a(n)) = 0.

Rémy Sigrist, Table of n, a(n) for n = 1..8190
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Repdigit

Subsequence of A154314.

Cf. A007089, A125289.

not[n_]:=Module[{c=DigitCount[n,3]},c[[1]]==0||c[[2]]==0]; Select[ Range[ 250],not] (* Harvey P. Dale, Dec 15 2012 *)

is(n, base=3) = #Set(select(sign, digits(n, base)))==1 \\ Rémy Sigrist, Mar 28 2020

a(n, base=3) = { for (w=0, oo, if (n<=(base-1)*2^w, my (d=1+(n-1)\2^w, k=2^w+(n-1)%(2^w)); return (d*fromdigits(binary(k), base)), n -= (base-1)*2^w)) } \\ Rémy Sigrist, Mar 28 2020

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

A212193 In ternary representation of n: a(n) = if n is pandigital then 3 else least digit not used.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, May 04 2012

a(A032924(n)) = 0; a(A081605(n)) <> 0;

a(A031944(n)) = 3; a(A154314(n)) <> 3.

			.   0 ->   '0':   a(0) = 1
.   1 ->   '1':   a(1) = 0
.   2 ->   '2':   a(2) = 0
.   3 ->  '10':   a(3) = 2
.   4 ->  '11':   a(4) = 0
.   5 ->  '12':   a(5) = 0
.   6 ->  '20':   a(6) = 1
.   7 ->  '21':   a(7) = 0
.   8 ->  '22':   a(8) = 0
.   9 -> '100':   a(9) = 2
.  10 -> '101':  a(10) = 2
.  11 -> '102':  a(11) = 3  <-- 11 is the smallest 3-pandigital number
.  12 -> '110':  a(12) = 2
.  13 -> '111':  a(13) = 0
.  14 -> '112':  a(14) = 0
.  15 -> '120':  a(15) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Pandigital Number
Wikipedia, Ternary numeral system
Wikipedia, Pandigital number

Cf. A007089, A067898 (decimal).

import Data.List (delete)
a212193 n = f n [0..3] where
   f x ys | x <= 2    = head $ delete x ys
          | otherwise = f x' $ delete d ys where (x',d) = divMod x 3

A043530 Number of distinct base-3 digits of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(A031944(n)) = 3; a(A154314(n)) < 3. - Reinhard Zumkeller, Jan 07 2009

Image, under the coding sending i -> 1+floor(i/3), of the fixed point, starting with 0, of the morphism 0 -> 012, 1 -> 314, 2 -> 542, 3 -> 336, 4 -> 644, 5 -> 565, 6 -> 666. - Jeffrey Shallit, May 15 2016

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A007089, A031944, A037897, A154314.

Table[Length[Union[IntegerDigits[n,3]]],{n,90}] (* Harvey P. Dale, May 13 2020 *)

a(n) = #vecsort(digits(n,3),,8); \\ Michel Marcus, May 16 2016

A239348 Numbers that are not pandigital in any base b >= 3.

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, 79, 80, 81, 82, 84, 85, 90, 91, 93, 94, 109, 111, 112, 117, 118, 121, 122, 124

Offset: 1

Joonas Pohjonen, Mar 16 2014

Identical to A154314 until a(51).

			11 is not in the sequence because 11 = 102_3.

Joonas Pohjonen, Table of n, a(n) for n = 1..10000

Cf. A154314.

nop[n_] := Block[{b=3, d}, While[ Length[d = IntegerDigits[n, b]] >= b &&  Union[d] != Range[0, b-1], b++]; Length[d] < b]; Select[Range[0, 124], nop] (* Giovanni Resta, Mar 17 2014 *)

isok(n) = {for (b = 3, n, d = digits(n, b); if (#vecsort(d,,8) == b, return(0));); return (1);} \\ Michel Marcus, Mar 17 2014

is(n)=for(b=3,log(n)\lambertw(log(n))+1, if(#Set(digits(n,b))==b, return(0))); 1 \\ Charles R Greathouse IV, Mar 17 2014

a(n) >> n^1.58..., where the exponent is log(3)/log(2). - Charles R Greathouse IV, Mar 17 2014

A239437 Least number that is pandigital in some base >= n but not pandigital in bases 3 through n-1.

Original entry on oeis.org

11, 78, 698, 12280, 179685, 5518135, 1037845296, 1037845296, 46935813565, 2860727439460, 285947759601954, 1018897102759406, 672654273047783383

Offset: 3

Charles R Greathouse IV, Mar 18 2014

Gives an upper bound on the testing needed to check membership in A239348.

Charles R Greathouse IV, GP script for computing terms

Subsequence of A154314. Cf. A239348.

PARI
```
See Greathouse link.
```

from itertools import permutations
from gmpy2 import digits
def A239437(n): # requires 3 <= n <= 62
    m = n
    while True:
        s = ''.join([digits(i,m) for i in range(m)])
        for d in permutations(s,m):
            if d[0] != '0':
                c = mpz(''.join(d),m)
                for b in range(3,n):
                    if len(set(digits(c,b))) == b:
                        break
                else:
                    return int(c)
        m += 1 # Chai Wah Wu, May 31 2015

Trivially a(n) >= A049363(n) > n^(n-1).

a(15) from Giovanni Resta, Mar 19 2014

cp's OEIS Frontend

A032924 Numbers whose ternary expansion contains no 0.

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A125292 Numbers having either no ones or no twos in their ternary representation.

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

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A212193 In ternary representation of n: a(n) = if n is pandigital then 3 else least digit not used.

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A043530 Number of distinct base-3 digits of n.

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A239348 Numbers that are not pandigital in any base b >= 3.

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A239437 Least number that is pandigital in some base >= n but not pandigital in bases 3 through n-1.

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