A081608 -id:A081608 - 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

A061392 a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Henry Bottomley, Apr 30 2001

Number of nonnegative integers < n having no 1 in their ternary representation. - Reinhard Zumkeller, Mar 23 2003; corrected by Henry Bottomley, Mar 24 2003

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
Wikipedia, Cantor function.

k appears A061393(k) times.
Cf. A007089, A062756, A081608, A081609, A081611.
Essentially the partial sums of A088917.

A061392 := proc(n) option remember; local a : if n <=1 then n else A061392(floor(n/3)) + A061392(ceil(n/3)) fi: end: seq(A061392(n), n=0..87); # Johannes W. Meijer, Jun 05 2011

a(n+1) + A081609(n) = n+1. - Reinhard Zumkeller, Mar 23 2003; corrected by Henry Bottomley, Mar 24 2003
From Johannes W. Meijer, Jun 05 2011: (Start)
a(3*n+1) = a(n+1) + a(n), a(3*n+2) = a(n+1) + a(n) and a(3*n+3) = 2*a(n+1), for n>=1, with a(0)=0, a(1)=1, a(2)=1 and a(3)=2. [Northshield]
G.f.: x*Product_{n>=0} (1 + x^(3^n) + 2*x^(2*3^n) + x^(3*3^n) + x^(4*3^n)). [Northshield] (End)
Apparently, for any n >= 0 and k such that n < 3^k, a(n) = 2^k * c(n / 3^k) where c is the Cantor function. - Rémy Sigrist, Jul 12 2019

A081607 Number of numbers <= n having at least one 0 in their ternary representation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 27, 27, 27, 28, 29, 30, 31, 31, 31, 32, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 45, 46, 46, 46, 47, 48, 49, 50

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A081608(n) = n+1.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A007089, A077267, A081609, A081610.

f:= n -> `if`(has(convert(n,base,3),0),1,0):
ListTools:-PartialSums(map(f, [$0..100])); # Robert Israel, Mar 18 2018

Accumulate[Boole[Table[DigitCount[n,3,0]>0,{n,0,80}]]] (* Harvey P. Dale, Jun 23 2017 *)

first(n)=my(s,t); vector(n,k, t=Set(digits(k,3)); s+=(t[1]==0)) \\ Charles R Greathouse IV, Sep 02 2015

A081611 Number of numbers <= n having no 2 in their ternary representation.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) is also the size of the subset of [1..n] when numbers are added greedily so as to not contain a 3-term arithmetic progression, i.e., according to A003278: a(n) = the largest k such that A003278(k) <= n. (Cf. A003002, the size of the optimal (largest) 3-free subset of [1..n].) - Shreevatsa R, Oct 19 2009

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

Cf. A003278, A007089, A039966, A081603, A081608, A061392, A081610.

a081611 n = a081611_list !! n
a081611_list = scanl1 (+) a039966_list
-- Reinhard Zumkeller, Jan 28 2012

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (&*[1+x^(3^k): k in [0..5]])/(1-x) )); // G. C. Greubel, Mar 29 2019

CoefficientList[Series[Product[1+x^(3^k), {k,0,5}]/(1-x), {x,0,100}], x] (* G. C. Greubel, Mar 29 2019 *)

{a(n)=local(A,m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*(1+x+x^2)*subst(A,x,x^3)); polcoeff(A,n))} /* Michael Somos, Aug 31 2006 */

first(n)=my(s,t); vector(n,k,t=Set(digits(k,3)); s+=t[#t]<2) \\ Charles R Greathouse IV, Sep 02 2015

my(x='x+O('x^100)); Vec(prod(k=0,5,1+x^(3^k))/(1-x)) \\ G. C. Greubel, Mar 29 2019

from gmpy2 import digits
def A081611(n):
    l = (s:=digits(n,3)).find('2')
    if l >= 0: s = s[:l]+'1'*(len(s)-l)
    return int(s,2)+1 # Chai Wah Wu, Dec 05 2024

(product(1+x^(3^k) for k in (0..5))/(1-x)).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Mar 29 2019

a(n) + A081610(n) = n+1.
From Michael Somos, Aug 31 2006: (Start)
G.f. A(x) satisfies A(x)=(1+x)*(1+x+x^2)*A(x^3).
G.f.: (1/(1-x))*Product_{k>=0} (1+x^(3^k)).
a(n) = a(n-1) + A039966(n). (End)

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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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A061392 a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.

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A081607 Number of numbers <= n having at least one 0 in their ternary representation.

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A081611 Number of numbers <= n having no 2 in their ternary representation.

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