A007089 -id:A007089 - cp's OEIS frontend

A360502 Concatenate the ternary strings for 1,2,...,n.

1, 12, 1210, 121011, 12101112, 1210111220, 121011122021, 12101112202122, 12101112202122100, 12101112202122100101, 12101112202122100101102, 12101112202122100101102110, 12101112202122100101102110111, 12101112202122100101102110111112, 12101112202122100101102110111112120

Offset: 1

N. J. A. Sloane, Feb 16 2023

If the terms are read as ternary strings and converted to base 10, we get A048435. For example, a(2) = 12_3 = 5_10, which is A048435(2). This is a prime, and gives the first term of A360503.
If the terms are read as decimal numbers, which of them are primes?  12101112202122100101102110111, for example, is not a prime, since it is 37*327057086543840543273030003.
When read as decimal numbers, the first prime is a(7315), with 56003 digits. - Michael S. Branicky, Apr 18 2023

			a(4): concatenate 1, 2, 10, 11, getting 121011.

Winston de Greef, Table of n, a(n) for n = 1..222
Index entries for sequences related to Most Wanted Primes video

Cf. A007089, A036954, A360503, A360504.

This is the ternary analog of A007908.

a:= proc(n) option remember; `if`(n=0, 0, (l-> parse(cat(
      a(n-1), seq(l[-i], i=1..nops(l)))))(convert(n, base, 3)))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 17 2023

nn = 15; s = IntegerDigits[Range[nn], 3]; Array[FromDigits[Join @@ s[[1 ;; #]]] &, nn] (* Michael De Vlieger, Apr 19 2023 *)

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(1, n+1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:=s+"".join(map(str, digits(n, 3)[1:]))) for n in count(1))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

A097580 Base 3 representation of the concatenation of the first n numbers with the most significant digits first.

Original entry on oeis.org

1, 110, 11120, 1200201, 121221020, 20021100110, 2022201111201, 212020020002100, 22121022020212200, 1011212101120110200001, 11101000122011021220211010, 121012010100112220022220220120

Offset: 1

Cino Hilliard, Aug 29 2004

Consider numbers of the form 1, 12, 123, 1234, ..., N. Find the highest power of 3^p such that 3^p <= N. Then p = [log(N)/log(3)] and for 0 <= qi <= 2 [N/3^p] = q1 + r1 [r1/3^(p-1)] = q2 + r2 ........................ rp/3^1 = qp + rp+1 rp+1/3^0 = qp+1 0 For N = 1234, p = [log(1234)/log(3)] = 6 division quot rem 1234/3^6 = 1 505 505/3^5 = 2 19 19/3^4 = 0 19 19/3^3 = 0 19 19/3^2 = 2 1 1/3^1 = 0 1 1/3^0 = 1 0 The sequence of quotients, top down, form the entry in the table for 1234. Obviously this algorithm works for any N.

			The 4th concatenation of the integers > 0 is 1234. base(10,3,1234) = 1200201 the 4th entry in the table.

Seiichi Manyama, Table of n, a(n) for n = 1..195

Cf. A007908, A050926, A097582, A097583.

Cf. A007089.

Table[FromDigits[IntegerDigits[FromDigits[Flatten[Table[ IntegerDigits[n],{n,i}]]],3]],{i,12}] (* Harvey P. Dale, May 23 2011 *)

a(n) = A007089(A007908(n)). - Seiichi Manyama, Apr 23 2022

A370859 Numbers m such that c(0) < c(1) > c(2), where c(k) = number of k's in the ternary representation of m.

Original entry on oeis.org

1, 4, 10, 12, 13, 14, 16, 22, 31, 32, 34, 37, 38, 39, 40, 41, 42, 43, 46, 48, 49, 58, 64, 66, 67, 85, 91, 93, 94, 95, 97, 103, 109, 111, 112, 113, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 129, 130, 131, 133, 139, 145, 147, 148, 149, 151, 157

Offset: 1

Clark Kimberling, Mar 03 2024

			The ternary representation of 16 is 121, for which c(0)=0 < c(1)=2 > c(2)=1.

Cf. A007089, A077267, A062756, A081603.

Cf. A355275, A370853, A370860, A370861, A370862, A370863, A370870.

filter:= proc(n) local L,m;
L:= convert(n,base,3); m:= numboccur(1,L);
numboccur(0,L) < m and numboccur(2,L) < m
end proc:
select(filter, [$1 .. 200]); # Robert Israel, Mar 03 2024

Select[Range[1000], DigitCount[#, 3, 0] < DigitCount[#, 3, 1] > DigitCount[#, 3, 2] &]

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

A081610 Number of numbers <= n having at least one 2 in their ternary representation.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 25, 25, 25, 26, 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

Offset: 0

Reinhard Zumkeller, Mar 23 2003

a(n) + A081611(n) = n+1. Partial sums of A189820.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A007089, A081603, A081607, A081609, A189820.

num2tern := proc(n) return numboccur(convert(n,base,3),2): end: a:=0: for n from 0 to 80 do a:=a+`if`(num2tern(n)>0,1,0): printf("%d, ",a): od: # Nathaniel Johnston, May 17 2011

Accumulate[Table[If[DigitCount[n,3,2]>0,1,0],{n,0,70}]] (* Harvey P. Dale, Aug 20 2012 *)

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

from gmpy2 import digits
def A081610(n):
    l = (s:=digits(n,3)).find('2')
    if l >= 0: s = s[:l]+'1'*(len(s)-l)
    return n-int(s,2) # Chai Wah Wu, Dec 05 2024

a(n) ~ n. - Charles R Greathouse IV, Sep 02 2015

A104320 Number of zeros in ternary representation of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 0, 4, 2, 3, 3, 3, 3, 3, 7, 7, 9, 5, 6, 6, 4, 4, 3, 5, 6, 7, 9, 9, 10, 6, 6, 9, 9, 8, 9, 8, 7, 13, 12, 13, 9, 5, 9, 8, 6, 16, 13, 9, 10, 11, 11, 7, 14, 13, 13, 9, 12, 14, 15, 15, 11, 11, 17, 15, 19, 14, 19, 12, 18, 15, 11, 10, 16, 15, 14, 14, 13, 17, 14

Offset: 0

Reinhard Zumkeller, Mar 01 2005

Conjecture from N. J. A. Sloane: a(n) > 0 for n > 15, see A102483.

			n=13: 2^13=8192 -> '102020102', a(13) = 4.

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

Cf. A004642, A007089.

[Multiplicity(Intseq(2^n,3),0):n in [0..90]]; // Marius A. Burtea, Nov 17 2019

f:= n -> numboccur(0, convert(2^n,base,3)):
map(f, [$0..100]); # Robert Israel, Nov 17 2019

Table[DigitCount[2^n,3,0],{n,0,90}] (* Harvey P. Dale, May 06 2014 *)

a(n) = my(d=vecsort(digits(2^n, 3))); #setintersect(d, vector(#d)) \\ Felix Fröhlich, Nov 17 2019

a(n) = #select(d->!d, digits(2^n, 3)); \\ Ruud H.G. van Tol, May 09 2024

a(n) = A077267(A000079(n)).

a(A104321(n))=n and a(m)<>n for m < A104321(n).

A154314 Numbers with not more than two distinct digits in ternary representation.

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, 78, 79, 80, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 122

Offset: 1

Reinhard Zumkeller, Jan 07 2009

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.

Complement of A031944.
Union of A032924, A005823 and A005836.
Cf. A007089, A043530, A212193.

import Data.List (findIndices)
a154314 n = a154314_list !! (n-1)
a154314_list = findIndices (/= 3) a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[0,200],Length[Union[IntegerDigits[#,3]]]<3&] (* Harvey P. Dale, Nov 23 2012 *)

is(n)=#Set(digits(n,3))<3 \\ Charles R Greathouse IV, Mar 17 2014

A043530(a(n)) <= 2.
A212193(a(n)) <> 3. - Reinhard Zumkeller, May 04 2012
a(n) >> n^1.58..., where the exponent is log(3)/log(2). - Charles R Greathouse IV, Mar 17 2014
Sum_{n>=2} 1/a(n) = 5.47555542241781419692840472181029603722178623821762258873485212626135391726959422416350447132335696748507... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A163325 Pick digits at the even distance from the least significant end of the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '102' (1*9 + 0*3 + 2), from which we pick the "zeroth" and 2nd digits from the right, giving '12' = 1*3 + 2 = 5, thus a(11) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..728
Kevin Ryde, Plot2 of X=A163325,Y=A163326, illustrating the ternary Z-order curve.
Index entries for sequences related to coordinates of 2D curves

A059905 is an analogous sequence for binary.

Cf. A007089, A163327, A163328, A163329.

a(n) = fromdigits(digits(n,9)%3,3); \\ Kevin Ryde, May 14 2020

a(0) = 0, a(n) = (n mod 3) + 3*a(floor(n/9)).
a(n) = Sum_{k>=0} {A030341(n,k)*b(k)} where b is the sequence (1,0,3,0,9,0,27,0,81,0,243,0... = A254006): powers of 3 alternating with zeros. - Philippe Deléham, Oct 22 2011
A037314(a(n)) + 3*A037314(A163326(n)) = n for all n.

Edited by Charles R Greathouse IV, Nov 01 2009

A163328 Square array A, where entry A(y,x) has the ternary digits of x interleaved with the ternary digits of y, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 9, 5, 7, 27, 10, 12, 8, 28, 30, 11, 13, 15, 29, 31, 33, 18, 14, 16, 36, 32, 34, 54, 19, 21, 17, 37, 39, 35, 55, 57, 20, 22, 24, 38, 40, 42, 56, 58, 60, 81, 23, 25, 45, 41, 43, 63, 59, 61, 243, 82, 84, 26, 46, 48, 44, 64, 66, 62, 244, 246, 83, 85, 87, 47, 49

Offset: 0

Antti Karttunen, Jul 29 2009

			From _Kevin Ryde_, Oct 06 2020: (Start)
Array A(y,x) read by downwards antidiagonals, so 0, 1,3, 2,4,6, etc.
        x=0   1   2   3   4   5   6   7   8
      +--------------------------------------
  y=0 |   0,  1,  2,  9, 10, 11, 18, 19, 20,
    1 |   3,  4,  5, 12, 13, 14, 21, 22,
    2 |   6,  7,  8, 15, 16, 17, 24,
    3 |  27, 28, 29, 36, 37, 38,
    4 |  30, 31, 32, 39, 40,
    5 |  33, 34, 35, 42,
    6 |  54, 55, 56,
    7 |  57, 58,
    8 |  60,
(End)

Antti Karttunen, Table of n, a(n) for n = 0..3320
Index entries for sequences that are permutations of the natural numbers

Inverse: A163329. Transpose: A163330. Cf. A037314 (row y=0), A208665 (column x=0)

Cf. A054238 is an analogous sequence for binary. Cf. A007089, A163327, A163332, A163334.

A(y,x) = 3*fromdigits(digits(y,3),9) + fromdigits(digits(x,3),9); \\ Kevin Ryde, Oct 06 2020

(define (A163328 n) (+ (A037314 (A025581 n)) (* 3 (A037314 (A002262 n)))))

a(n) = A037314(A025581(n)) + 3*A037314(A002262(n))

a(n) = A163327(A163330(n)).

Edited by Charles R Greathouse IV, Nov 01 2009

A171960 Values of the 2-complement of n in ternary representation.

Original entry on oeis.org

2, 1, 0, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2

Offset: 0

Reinhard Zumkeller, Jan 20 2010

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

Cf. A003462, A007089, A035327, A061601, A134025, A134026.

a[n_] := 3^(1 + Floor[Log[3, n]]) - n - 1; a[0] = 2; Array[a, 100] (* Amiram Eldar, Apr 03 2025 *)

a(n) = 3^(if(n,logint(n,3))+1) - 1 - n; \\ Kevin Ryde, Jul 16 2020

a(n) = if n < 3 then 2 - n else 3*a(floor(n/3)) + 2 - n mod 3.
a(A134026(n)) < A134026(n).
a(A003462(n)) = A003462(n).
a(A134025(n)) >= A134025(n).

cp's OEIS Frontend

A360502 Concatenate the ternary strings for 1,2,...,n.

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A097580 Base 3 representation of the concatenation of the first n numbers with the most significant digits first.

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A370859 Numbers m such that c(0) < c(1) > c(2), where c(k) = number of k's in the ternary representation of m.

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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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A081610 Number of numbers <= n having at least one 2 in their ternary representation.

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A104320 Number of zeros in ternary representation of 2^n.

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A154314 Numbers with not more than two distinct digits in ternary representation.

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