A081605 -id:A081605 - 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

A077267 Number of zeros in base-3 expansion 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, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 2

Offset: 0

Henry Bottomley, Nov 01 2002

			a(8)=0 since 8 written in base 3 is 22 with 0 zeros;
a(9)=2 since 9 written in base 3 is 100 with 2 zeros;
a(10)=1 since 10 written in base 3 is 101 with 1 zero.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters, F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A023416, A077266, A062756, A081603.

Cf. A007089, A081605, A032924, A081607, A081608, A077267, A134023. - Reinhard Zumkeller, Mar 23 2003

a077267 n = a079978 n + if n < 3 then 0 else a077267 (n `div` 3)
-- Reinhard Zumkeller, Feb 21 2013

Table[Count[IntegerDigits[n,3],0],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Jul 25 2009 *)
DigitCount[Range[0,110],3,0] (* Harvey P. Dale, Jul 04 2021 *)

a(1)=a(2)=0; a(3n)=a(n)+1; a(3n+1)=a(3n+2)=a(n). a(3^n-2)=a(3^n-1)=0; a(3^n)=n. a(n)=A077266(n, 3).
a(n) + A062756(n) + A081603(n) = A081604(n). - Reinhard Zumkeller, Mar 23 2003
G.f.: (Sum_{k>=0} x^(3^(k+1))/(1 + x^(3^k) + x^(2*3^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A079978(n) if n < 3, A079978(n) + a(floor(n/3)) otherwise. - Reinhard Zumkeller, Feb 21 2013

a(0)=1 added, offset changed to 0 and b-file adjusted by Reinhard Zumkeller, Feb 21 2013

Wrong formula deleted by Reinhard Zumkeller, Feb 21 2013

A074940 Numbers having at least one 2 in their ternary representation.

Original entry on oeis.org

2, 5, 6, 7, 8, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 32, 33, 34, 35, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 86, 87, 88, 89, 92

Offset: 1

Benoit Cloitre and Reinhard Zumkeller, Oct 04 2002; revised Dec 03 2003

Also, numbers m such that 3 divides C(2m,m).
Also, numbers m such that the central trinomial coefficient A002426(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Also, numbers m such that A092255(m) == 0 (mod 3). - Benoit Cloitre, Mar 22 2004
Also, numbers m such that the coefficient of x^m equals 0 in Product_{k>=0} (1-x^(3^k)). - N. J. A. Sloane, Jun 01 2010

			12 is not in the sequence since it is 110_3, but 11 is in the sequence since it is 102_3. - _Michael B. Porter_, Jun 30 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Complement of A005836.
Cf. A006996, A007089, A081603, A081610, A081605, A081606.
A039966(a(n)) = 0.

a074940 n = a074940_list !! (n-1)
a074940_list = filter ((== 0) . a039966) [0..]
-- Reinhard Zumkeller, Jun 06 2012, Sep 29 2011

Select[Range@ 120, MemberQ[IntegerDigits[#, 3], 2] &] (* or *)
Select[Range@ 120, Divisible[Binomial[2 #, #], 3] &] (* Michael De Vlieger, Jun 29 2016 *)
Select[Range[100],DigitCount[#,3,2]>0&] (* Harvey P. Dale, Aug 25 2019 *)

is(n)=while(n,if(n%3==2,return(1));n\=3);0 \\ Charles R Greathouse IV, Aug 21 2011

from gmpy2 import digits
def A074940(n):
    def f(x):
        s = digits(x,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n+int(s[:i]+'1'*(l-i),2)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 29 2024

a(n) = n + O(n^0.631). - Charles R Greathouse IV, Aug 21 2011

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A081606 Numbers having at least one 1 in their ternary representation.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A005823.
Integers m such that central Delannoy number A001850(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Integers m such that A026375(m) == 0 (mod 3). - Fabio Visonà, Aug 03 2023

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p.25.
Mathematics Stack Exchange, Proof that integers m in this sequence are such that A026375(m) == 0 (mod 3).

Cf. A007089, A062756, A081609, A081605, A074940.

Select[Range[100],DigitCount[#,3,1]>0&] (* Harvey P. Dale, Nov 26 2022 *)

from itertools import count, islice
def A081606_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = int(bin(n)[2:],3)<<1
        yield from range(a+1,b)
        a = b
A081606_list = list(islice(A081606_gen(),30)) # Chai Wah Wu, Oct 13 2023

from gmpy2 import digits
def A081606(n):
    def f(x):
        s = digits(x>>1,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n-1+(int(s[:i] or '0',2)+1<Chai Wah Wu, Oct 29 2024

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

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

A382413 Numbers with at least one zero in their base-7 representation.

Original entry on oeis.org

0, 7, 14, 21, 28, 35, 42, 49, 50, 51, 52, 53, 54, 55, 56, 63, 70, 77, 84, 91, 98, 99, 100, 101, 102, 103, 104, 105, 112, 119, 126, 133, 140, 147, 148, 149, 150, 151, 152, 153, 154, 161, 168, 175, 182, 189, 196, 197, 198, 199, 200, 201, 202, 203, 210, 217, 224, 231, 238

Offset: 1

Paolo Xausa, Mar 24 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007093, A043393, A382412 (complement).

Select[Range[0, 250], DigitCount[#, 7, 0] > 0 &]

A370916 Positive integers whose ternary representation includes at least one 0, and every 0 is followed by 1.

Original entry on oeis.org

10, 19, 31, 32, 37, 46, 58, 59, 64, 73, 91, 94, 95, 97, 98, 112, 113, 118, 127, 139, 140, 145, 154, 172, 175, 176, 178, 179, 193, 194, 199, 208, 220, 221, 226, 235, 274, 275, 280, 283, 284, 286, 287, 289, 292, 293, 295, 296, 334, 337, 338, 340, 341, 355, 356

Offset: 1

Clark Kimberling, Mar 13 2024

			The ternary representations of 10, 19, and 31 are 101, 201, and 1011.

Cf. A007089, A081605.

Cf. A370917, A370918, A370919, A370920, A370921, A370922, A370923.

Map[#[[1]] &, Select[Map[{#, #[[1]] > 0 && #[[1]] == #[[2]] &[{Length[
StringCases[#, "0"]], Length[StringCases[#, "01"]]}] &[
IntegerString[#, 3]]} &, Range[500]], #[[2]] &]]
 (* Peter J. C. Moses, Mar 05 2024 *)

A370917 Positive integers whose ternary representation includes at least one 0, and every 0 is followed by 2.

Original entry on oeis.org

11, 20, 34, 35, 38, 47, 61, 62, 65, 74, 101, 103, 104, 106, 107, 115, 116, 119, 128, 142, 143, 146, 155, 182, 184, 185, 187, 188, 196, 197, 200, 209, 223, 224, 227, 236, 304, 305, 308, 310, 311, 313, 314, 317, 319, 320, 322, 323, 344, 346, 347, 349, 350, 358

Offset: 1

Clark Kimberling, Mar 15 2024

			The ternary representations of 11, 20, and 34 are 102, 202, and 1021.

Cf. A007089, A081605.

Cf. A370916, A370918, A370919, A370920, A370921, A370922, A370923.

Map[#[[1]] &, Select[Map[{#, #[[1]] > 0 && #[[1]] == #[[2]] &[{Length[
StringCases[#, "0"]], Length[StringCases[#, "02"]]}] &[
IntegerString[#, 3]]} &, Range[500]], #[[2]] &]]
  (* Peter J. C. Moses, Mar 05 2024 *)

A382415 Numbers with at least one zero in their base-5 representation.

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 26, 27, 28, 29, 30, 35, 40, 45, 50, 51, 52, 53, 54, 55, 60, 65, 70, 75, 76, 77, 78, 79, 80, 85, 90, 95, 100, 101, 102, 103, 104, 105, 110, 115, 120, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382416 (base 6), A382413 (base 7), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007091, A023721 (complement), A023722.

Select[Range[0, 150], DigitCount[#, 5, 0] > 0 &]

A382416 Numbers with at least one zero in their base-6 representation.

Original entry on oeis.org

0, 6, 12, 18, 24, 30, 36, 37, 38, 39, 40, 41, 42, 48, 54, 60, 66, 72, 73, 74, 75, 76, 77, 78, 84, 90, 96, 102, 108, 109, 110, 111, 112, 113, 114, 120, 126, 132, 138, 144, 145, 146, 147, 148, 149, 150, 156, 162, 168, 174, 180, 181, 182, 183, 184, 185, 186, 192, 198

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382413 (base 7), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007092, A043369, A248910 (complement).

Select[Range[0, 200], DigitCount[#, 6, 0] > 0 &]

cp's OEIS Frontend

A032924 Numbers whose ternary expansion contains no 0.

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A077267 Number of zeros in base-3 expansion of n.

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A074940 Numbers having at least one 2 in their ternary representation.

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A081606 Numbers having at least one 1 in their ternary representation.

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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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A382413 Numbers with at least one zero in their base-7 representation.

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A370916 Positive integers whose ternary representation includes at least one 0, and every 0 is followed by 1.

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