A059852 -id:A059852 - 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

A060109 Numbers in Morse code, with 1 for a dot, 2 for a dash and 0 between digits/letters.

Original entry on oeis.org

22222, 12222, 11222, 11122, 11112, 11111, 21111, 22111, 22211, 22221, 12222022222, 12222012222, 12222011222, 12222011122, 12222011112, 12222011111, 12222021111, 12222022111, 12222022211, 12222022221, 11222022222, 11222012222, 11222011222, 11222011122, 11222011112, 11222011111

Offset: 0

Henry Bottomley, Feb 28 2001

			a(10) = 12222022222 since 1 is ".----" and 0 is "-----".

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

Cf. A059852 (Morse code for letters), A008777 (number of dots and dashes).

Cf. A060110 (these base-3 numbers converted to decimal), A321332 (duration of the code for n).

import Data.List (inits, tails)
a060109 n = if n == 0 then 22222 else read (conv n) :: Integer where
   conv 0 = []
   conv x = conv x' ++ mCode !! d where (x', d) = divMod x 10
   mCode = map ('0' :) (mc ++ (reverse $ init $ tail $ map reverse mc))
   mc = zipWith (++) (inits "111111") (tails "22222")
-- Reinhard Zumkeller, Feb 20 2015

With[{a = Association@ Array[# -> If[# < 6, PadRight[ConstantArray[1, #], 5, 2], PadRight[ConstantArray[2, # - 5], 5, 1]] &, 10, 0]}, Array[FromDigits@ Flatten@ Riffle[Map[Lookup[a, #] &, IntegerDigits[#]], 0] &, 25]] (* Michael De Vlieger, Nov 02 2020 *)

apply( {A060109(n)=if(n>9,self()(n\10)*10^6)+fromdigits([1+(abs(k-n%10)>2)|k<-[3..7]])}, [0..39]) \\ M. F. Hasler, Jun 22 2020

a(n) A007089(A060110(n)) = a(floor(n/10))*10^6 + a(n%10) for n > 9 and a(n) = 33333 - a(n-5) for n%10 > 4, where % is the modulo (remainder) operator. - M. F. Hasler, Jun 22 2020

A008777 Number of dots and dashes used when representing n-th letter in Morse code.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Number of digits of A059852 written in base 3.

Cf. A059852 (Morse code for letters A-Z, read in base 3).

apply( {A008777(n)=logint(A059852[n],3)+1}, [1..26]) \\ assuming that A059852 is a vector rather than a function. - M. F. Hasler, Jun 22 2020

a(n) = floor(log_3(A059852(n))) + 1. - M. F. Hasler, Jun 22 2020

A060110 Numbers in Morse code, with 1 for a dot, 2 for a dash and 0 between digits/letters and then converted from base 3 to base 10.

Original entry on oeis.org

242, 161, 134, 125, 122, 121, 202, 229, 238, 241, 117611, 117530, 117503, 117494, 117491, 117490, 117571, 117598, 117607, 117610, 97928, 97847, 97820, 97811, 97808, 97807, 97888, 97915, 97924, 97927, 91367, 91286, 91259, 91250, 91247

Offset: 0

Henry Bottomley, Feb 28 2001

The mentioned base 3 representation of the terms (digits 0, 1 and 2) is given in A060109. - M. F. Hasler, Jun 22 2020

			a(10) = 12222022222[base 3] = 117611[base 10] since 1 is ".----" and 0 is "-----".

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

Cf. A008777, A059852, A060109.

Cf. A007089.

a060110 = t . a060109 where
   t 0 = 0
   t n = if n == 0 then 0 else 3 * t n' + d  where (n', d) = divMod n 10
-- Reinhard Zumkeller, Feb 20 2015

apply( {A060110(n)=if(n>9, self()(n\10)*3^6)+fromdigits([1+(abs(k-n%10)>2)|k<-[3..7]],3)}, [0..39]) \\ M. F. Hasler, Jun 23 2020

A007089(a(n)) = A060109(n). - Reinhard Zumkeller, Feb 20 2015

A105386 Morse code alphabet where "." = 0 and "-" = 1, binary converted to decimal.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 02 2005

Hamming weight A000120(a(n)) gives number of dashes in the Morse code of the n-th letter. A008777 allows us to know how many leading zeros a term should have. A059852 is a better (unambiguous) variant. - M. F. Hasler, Jun 23 2020

			a(3) = 'C' = "-.-." -> 1010 (binary) = 10.

Omniglot, Morse Code

Cf. A105387, A059852 (Morse alphabet), A000120 (binary weight), A008777 (number of dots and dashes in Morse code for n-th letter).

A105386=[fromdigits([d-1|d<-digits(t,3)],2)|t<-A059852] \\ M. F. Hasler, Jun 23 2020

Name and crossrefs edited by M. F. Hasler, Jun 23 2020

A105387 Morse code alphabet where "-" = 0 and "." = 1, binary converted to decimal.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 02 2005

			a(3) = 'C' = "-.-." -> 0101 (binary) = 5.

Omniglot, Morse Code

Cf. A105386, A059852 (Morse alphabet), A000120 (binary weight), A008777 (number of dots and dashes in Morse code for n-th letter).

A105387=[fromdigits([2-d|d<-digits(t,3)],2)|t<-A059852] \\ M. F. Hasler, Jun 23 2020

Name and crossrefs edited by M. F. Hasler, Jun 23 2020

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A032924 Numbers whose ternary expansion contains no 0.

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A060109 Numbers in Morse code, with 1 for a dot, 2 for a dash and 0 between digits/letters.

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A008777 Number of dots and dashes used when representing n-th letter in Morse code.

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A060110 Numbers in Morse code, with 1 for a dot, 2 for a dash and 0 between digits/letters and then converted from base 3 to base 10.

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A105386 Morse code alphabet where "." = 0 and "-" = 1, binary converted to decimal.

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A105387 Morse code alphabet where "-" = 0 and "." = 1, binary converted to decimal.

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