A032924 -id:A032924 - cp's OEIS frontend

A132140 Numbers containing no zeros in ternary representation and with an initial 1.

1, 4, 5, 13, 14, 16, 17, 40, 41, 43, 44, 49, 50, 52, 53, 121, 122, 124, 125, 130, 131, 133, 134, 148, 149, 151, 152, 157, 158, 160, 161, 364, 365, 367, 368, 373, 374, 376, 377, 391, 392, 394, 395, 400, 401, 403, 404, 445, 446, 448, 449, 454, 455, 457, 458, 472

Offset: 1

Reinhard Zumkeller, Aug 20 2007

Intersection of A032924 and A132141;

A132138(A132139(a(n))) = 1.

			14 is in the sequence because its ternary representation is 112;
15 is not in the sequence because its ternary representation is 120.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 3-automatic sequences.

Cf. A032924, A132141, A132138, A132139, A007089.

a132140 n = a132140_list !! (n-1)
a132140_list = filter f [1..] where
   f x = x < 3 && x == 1 || t > 0 && f x' where (x', t) = divMod x 3
-- Reinhard Zumkeller, Feb 06 2015

a:=proc(n) local nn, L: nn:=convert(n,base,3): L:=nops(nn): if nn[L]=1 and 0 < product(nn[j],j=1..L) then n else end if end proc: seq(a(n),n=1..500); # Emeric Deutsch, Sep 09 2007

Flatten[Table[FromDigits[Join[{1},#],3]&/@Tuples[{1,2},n],{n,0,5}]] (* Harvey P. Dale, Jan 28 2015 *)

A082555 Primes whose base-3 representation does not contain a 0.

Original entry on oeis.org

2, 5, 7, 13, 17, 23, 41, 43, 53, 67, 71, 79, 131, 149, 151, 157, 211, 229, 233, 239, 241, 367, 373, 401, 449, 457, 607, 617, 619, 643, 647, 691, 701, 719, 727, 1093, 1097, 1103, 1123, 1129, 1187, 1201, 1213, 1367, 1373, 1427, 1429, 1447, 1453, 1823, 1831, 1861

Offset: 1

Randy L. Ekl, May 03 2003

Primes in A032924. - Robert Israel, Dec 28 2018

The analog "primes without digit 2 in ternary" is A077717. There is no prime > 2 not having the digit 1 in ternary, since then the number is divisible by 2. - M. F. Hasler, Feb 15 2023

			41 = 1112_3, which contains no 0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A032924 (numbers without digit 0 in base 3), A073779, A077267.

Cf. A077717 (primes that are the sum of distinct powers of 3 <=> base-3 representation does not contain a digit 2).

select(t -> isprime(t) and not(has(convert(t,base,3),0)), [2,seq(i,i=5..10000,2)]); # Robert Israel, Dec 28 2018

dec3(s)=while(s>0,if(s%3==0,return(0),s=floor(s/3))); return(1)
forprime(i=1,20000,if(dec3(i)==1,print1(i,", "),))

def is_A082555(n): return is_A032924(n) and A010051(n)
[p for p in range(1888) if is_A082555(p)] # M. F. Hasler, Feb 15 2023

A106370 Smallest b > 1 such that n contains no zeros in its base b representation.

Original entry on oeis.org

2, 3, 2, 3, 3, 4, 2, 3, 4, 4, 4, 5, 3, 3, 2, 3, 3, 5, 5, 6, 4, 3, 3, 5, 3, 3, 4, 6, 4, 4, 2, 5, 5, 5, 6, 5, 4, 4, 4, 3, 3, 4, 3, 3, 4, 4, 4, 5, 3, 3, 6, 3, 3, 4, 4, 5, 4, 4, 4, 7, 4, 4, 2, 5, 6, 5, 3, 3, 5, 3, 3, 5, 5, 5, 7, 3, 3, 7, 3, 3, 5, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 5, 5, 6, 4, 4, 4, 6, 4

Offset: 1

Reinhard Zumkeller, May 01 2005

			n = 20: 20[binary] = '101001', 20[ternary] = '202', 20[base-4] = '110', 20[base-5] = '40', all containing at least one zero, but: 20[base-6] = '32', containing no zero therefore a(20) = 6.

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

Cf. A106371.

Cf. A000225, A032924, A106372, A119352.

a106370 n = f 2 n where
   f b x = g x where
     g 0 = b
     g z = if r == 0 then f (b + 1) n else g z'
           where (z', r) = divMod z b
-- Reinhard Zumkeller, Apr 12 2015

a[n_] := Module[{b = 2}, While[MemberQ[IntegerDigits[n, b], 0], b++]; b]; Array[a, 100] (* Amiram Eldar, Jul 29 2025 *)

a(n*a(n)+k) <= a(n) for 1 <= k < a(n).
a(A106372(n)) = n and a(m) <> n for m < A106372(n).
a(A000225(n)) = 2; a(A032924(n)) = 3 for n <> 5.

Typo in comment fixed by Reinhard Zumkeller, Aug 06 2010

A107680 Repeating k-th ternary repunit (A003462) 2^k times, k >= 0.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 4, 13, 13, 13, 13, 13, 13, 13, 13, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 0

Reinhard Zumkeller, May 20 2005

nonn

a(n) is the greatest ternary repunit that is not greater than the n-th number with no 2 in ternary representation.

			k=1: A003462(1) = (3^1-1)/2 = 1, therefore a(1) = a(2^1) = 1;
k=2: A003462(2) = (3^2-1)/2 = 4, therefore a(2+1) = a(2+2) =
a(2+3) = a(2+2^2) = 4.

Cf. A007089, A003462 (repunits in base 3), A000523 (number of digits in binary representation of n).

With[{nn=5},Flatten[Table[#[[1]],{#[[2]]}]&/@Thread[{Table[FromDigits[ PadRight[{},n,1],3],{n,nn}],2^Range[nn]}]]] (* Harvey P. Dale, Jan 04 2013 *)

apply( {A107680(n)=3^exponent(n+1)\2}, [0..66]) \\ M. F. Hasler, Jun 22 2020

def A107680(n): return 3**((n+1).bit_length()-1)-1>>1 # Chai Wah Wu, Nov 07 2024

A032924(n) = a(n) + A107681(n);
A081604(A107681(n)) <= A081604(a(n)) = A081604(A032924(n)) = A000523(n+1).
a(n) = A003462(A000523(n+1)).

Corrected by T. D. Noe, Oct 25 2006

Extended to a(0) = 0 by M. F. Hasler, Jun 23 2020

A107681 Repeat(first 2^k numbers with no 2 in ternary representation) for k>0.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 0, 1, 3, 4, 9, 10, 12, 13, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37

Offset: 1

Reinhard Zumkeller, May 20 2005

let A032924(n) = Sum(d(i)*3^i: 0

then .... a(n) = Sum((d(i)-1)*3^i: 0<=i

A032924(n) = A107680(n) + a(n);

A081604 (a(n)) <= A081604(A107680(n)) = A081604(A032924(n)) = A000523(n+1).

			A032924(177) = A107680(177) + a(177),
....... 1420 = ....... 1093 + 327,
.. '1221121' = ... '1111111'+ '110010',
............ = . A003462(7) + A005836(51).

Ruud H.G. van Tol, Table of n, a(n) for n = 1..8190 (12 rows).

Cf. A000523, A005836, A003462, A007089, A032924, A062050, A081604, A107680.

a(n)= fromdigits(binary(n+1-1<Ruud H.G. van Tol, Nov 18 2024

def A107681(n): return int(bin(n+1)[3:],3) # Chai Wah Wu, May 06 2025

a(n) = A005836(A062050(n+1)).

Data corrected by Ruud H.G. van Tol, Nov 18 2024.

A330166 Length of the longest run of 0's in the ternary expression of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Joshua Oliver, Dec 04 2019

All numbers appear in this sequence. The n-th power of 3 (A000244(n)) has n 0's in its ternary expression.

The longest run of zeros possible in this sequence is 2, as the last digit of the ternary expression of the integers cycles between 0, 1, and 2, meaning that at least one of three consecutive numbers has a 0 in its ternary expression.

			For n = 87, the ternary expression of 87 is 10020. The length of the runs of 0's in the ternary expression of 87 are 2 and 1, respectively. The larger of these two values is 2, so a(87) = 2.
   n [ternary n] a(n)
   0 [        0] 1
   1 [        1] 0
   2 [        2] 0
   3 [      1 0] 1
   4 [      1 1] 0
   5 [      1 2] 0
   6 [      2 0] 1
   7 [      2 1] 0
   8 [      2 2] 0
   9 [    1 0 0] 2
  10 [    1 0 1] 1
  11 [    1 0 2] 1
  12 [    1 1 0] 1
  13 [    1 1 1] 0
  14 [    1 1 2] 0
  15 [    1 2 0] 1
  16 [    1 2 1] 0
  17 [    1 2 2] 0
  18 [    2 0 0] 2
  19 [    2 0 1] 1
  20 [    2 0 2] 1

Wikipedia, Ternary numeral system.

Cf. A000244, A007089, A077267, A087117, A330036, A330167, A330168.

Equals zero iff n is in A032924.

Table[Max@FoldList[If[#2==0,#1+1,0]&,0,IntegerDigits[n,3]],{n,0,90}]

a(A000244(n)) = a(3^n) = n.

a(n) = 0 iff n is in A032924.

A360413 Irregular table T(n, k), n >= 0, k = 1..A002487(n+1), read by rows; the n-th row lists the numbers k such that A065361(k) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 7, 10, 8, 11, 12, 13, 14, 15, 18, 27, 16, 19, 28, 17, 20, 21, 29, 30, 22, 31, 23, 24, 32, 33, 36, 25, 34, 37, 26, 35, 38, 39, 40, 41, 42, 45, 54, 81, 43, 46, 55, 82, 44, 47, 48, 56, 57, 83, 84, 49, 58, 85, 50, 51, 59, 60, 63, 86, 87, 90

Offset: 0

Rémy Sigrist, Feb 06 2023

As a flat sequence, this is a permutation of the nonnegative integers with inverse A360414.

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   0  0
   1  1
   2  2, 3
   3  4
   4  5, 6, 9
   5  7, 10
   6  8, 11, 12
   7  13
   8  14, 15, 18, 27
   9  16, 19, 28
  10  17, 20, 21, 29, 30
  11  22, 31
  12  23, 24, 32, 33, 36
.
Table T(n, k) begins (with terms given in base 3):
  n   n-th row in base 3
  --  -------------------------
   0  0
   1  1
   2  2, 10
   3  11
   4  12, 20, 100
   5  21, 101
   6  22, 102, 110
   7  111
   8  112, 120, 200, 1000
   9  121, 201, 1001
  10  122, 202, 210, 1002, 1010
  11  211, 1011
  12  212, 220, 1012, 1020, 1100

Rémy Sigrist, Table of n, a(n) for n = 0..9841
Rémy Sigrist, PARI program
Index entries for sequences related to Stern's sequences
Index entries for sequences that are permutations of the natural numbers

Cf. A002487, A005836, A032924, A065361, A360414 (inverse).

PARI
```
See Links section.
```

T(n, 1) = A032924(n) for any n > 0.
T(n, A002487(n+1)) = A005836(n+1).
A065361(T(n, k)) = n.

A365278 In the binary expansion of n replace each run of k consecutive 1's by the decimal digits of A007931(k) to get the ternary expansion of a(n).

Original entry on oeis.org

0, 1, 3, 2, 9, 10, 6, 4, 27, 28, 30, 11, 18, 19, 12, 5, 81, 82, 84, 29, 90, 91, 33, 31, 54, 55, 57, 20, 36, 37, 15, 7, 243, 244, 246, 83, 252, 253, 87, 85, 270, 271, 273, 92, 99, 100, 93, 32, 162, 163, 165, 56, 171, 172, 60, 58, 108, 109, 111, 38, 45, 46, 21

Offset: 0

Rémy Sigrist, Aug 30 2023

This sequence is a permutation of the nonnegative integers with inverse A365279.
For any pair (b, c) of bases >= 2, we can devise a similar sequence, say F_{b, c}:
- for any d >= 2, let Z_d be the set of zeroless numbers in base d,
- in the base b expansion of n replace each run of consecutive nonzero digits (say corresponding to Z_b(k) for some k > 0) by the base c digits of Z_c(k) to get the base c expansion of F_{b, c}(n),
- F_{b, c} is a permutation of the nonnegative integers with inverse F_{c, b},
- F_{c, d} o F_{b, c} = F_{b, d} and F_{b, b} is the identity,
- in particular the present sequence corresponds to F_{2, 3} and its inverse to F_{3, 2}.

			The binary expansion of 415 is "110011111", A007931(2) = 2 and A007931(5) = 21, so the ternary expansion of a(415) is "20021", and a(415) = 169.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A007931, A023416, A032924, A077267, A290308, A365279 (inverse).

A007931[n_]:=Rest[IntegerDigits[n+1,2]]+1;
A365278[n_]:=FromDigits[Flatten[Map[If[First[#]==1,A007931[Length[#]],#]&,Split[IntegerDigits[n,2]]]],3];
Array[A365278,100,0] (* Paolo Xausa, Oct 17 2023 *)

PARI
```
See Links section.
```

a(2*n) = 3*a(n).
a(2^k - 1) = A032924(k) for any k > 0.
A077267(a(n)) = A023416(n).

A191109 a(1)=1, and if x is a term then 3x-1 and 3x+2 are terms too.

Original entry on oeis.org

1, 2, 5, 8, 14, 17, 23, 26, 41, 44, 50, 53, 68, 71, 77, 80, 122, 125, 131, 134, 149, 152, 158, 161, 203, 206, 212, 215, 230, 233, 239, 242, 365, 368, 374, 377, 392, 395, 401, 404, 446, 449, 455, 458, 473, 476, 482, 485, 608, 611, 617, 620, 635, 638, 644, 647, 689, 692, 698, 701, 716, 719, 725, 728, 1094, 1097, 1103, 1106, 1121

Offset: 1

Clark Kimberling, May 26 2011

nonn

See discussions at A190803, A191106.
The positive integers in (1+A191109)/3 comprise A153775, a proper subsequence of A191109.
The positive integers in (-2+A191109)/3 comprise A032924, a proper subsequence of A191109.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Juha Honkala, On number systems with negative digits, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, Vol. 14, 1989, pp. 149-156.

Cf. A190803, A191106, A153775, A032924.

h = 3; i = -1; j = 3; k = 2; f = 1;  g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* 191109 *)
b = (a + 1)/3; c = (a - 2)/3; r = Range[1, 900];
d = Intersection[b, r] (* A153775 *)
e = Intersection[c, r] (* A032924 *)
Nest[Flatten[{#,3#-1,3#+2}]&,1,10]//Union (* Harvey P. Dale, Apr 05 2020 *)

Name edited by Michel Marcus, Jul 29 2021

A360415 a(n) is the greatest number k not yet in the sequence such that A065361(n) = A065361(k).

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 6, 10, 12, 5, 7, 11, 8, 13, 27, 18, 28, 30, 15, 19, 29, 21, 31, 36, 33, 37, 39, 14, 16, 20, 17, 22, 32, 24, 34, 38, 23, 25, 35, 26, 40, 81, 54, 82, 84, 45, 55, 83, 57, 85, 90, 87, 91, 93, 42, 46, 56, 48, 58, 86, 63, 88, 92, 60, 64, 89, 66, 94

Offset: 0

Rémy Sigrist, Feb 06 2023

This sequence is a self-inverse permutation of the nonnegative integers.

			There are four numbers k such that A065361(k) = 8: 14, 15, 18, 27,
- so a(14) = 27,
     a(15) = 18,
     a(18) = 15,
     a(27) = 14.

Rémy Sigrist, Table of n, a(n) for n = 0..9841
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A360434 for a similar sequence.

Cf. A002487, A005836, A032924, A065361, A360413.

PARI
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See Links section.
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a(A360413(n, k)) = A360413(n, A002487(n) + 1 - k).
a(A032924(n)) = A005836(n+1) for any n > 0.
a(A005836(n+1)) = A032924(n) for any n > 0.

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A132140 Numbers containing no zeros in ternary representation and with an initial 1.

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A082555 Primes whose base-3 representation does not contain a 0.

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A106370 Smallest b > 1 such that n contains no zeros in its base b representation.

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A107680 Repeating k-th ternary repunit (A003462) 2^k times, k >= 0.

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A107681 Repeat(first 2^k numbers with no 2 in ternary representation) for k>0.

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A330166 Length of the longest run of 0's in the ternary expression of n.

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A360413 Irregular table T(n, k), n >= 0, k = 1..A002487(n+1), read by rows; the n-th row lists the numbers k such that A065361(k) = n.

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