A134026 -id:A134026 - cp's OEIS frontend

A081604 Number of digits in ternary representation of n.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Mar 23 2003

a(n) is the length of row n in table A054635. - Reinhard Zumkeller, Sep 05 2014

			a(8) = 2 because 8 = 22_3, having 2 digits.
a(9) = 3 because 9 = 100_3, having 3 digits.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary.

Cf. A003137, A007089, A030341, A049803, A054635, A062153, A070939, A080342.
Cf. A062756, A077267, A081603.
Cf. A134021, A134025, A134026, A246435.

a081604 n = if n < 3 then 1 else a081604 (div n 3) + 1
-- Reinhard Zumkeller, Sep 05 2014, Feb 21 2013

A081604 := proc(n)
    max(1,1+ilog[3](n)) ;
end proc: # R. J. Mathar, Jul 12 2016

Table[Length[IntegerDigits[n, 3]], {n, 0, 99}] (* Alonso del Arte, Dec 30 2012 *)
Join[{1},IntegerLength[Range[120],3]] (* Harvey P. Dale, Apr 07 2019 *)

a(n) = A062153(n) + 1 for n >= 1.
a(n) = A077267(n) + A062756(n) + A081603(n);
From Reinhard Zumkeller, Oct 19 2007: (Start)
0 <= A134021(n) - a(n) <= 1;
a(A134025(n)) = A134021(A134025(n));
a(A134026(n)) = A134021(A134026(n)) - 1. (End)
a(n+1) = -Sum_{k=1..n} mu(3*k)*floor(n/k). - Benoit Cloitre, Oct 21 2009
a(n) = floor(log_3(n)) + 1. - Can Atilgan and Murat Erşen Berberler, Dec 05 2012
a(n) = if n < 3 then 1 else a(floor(n/3)) + 1. - Reinhard Zumkeller, Sep 05 2014
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(3^k). - Ilya Gutkovskiy, Jan 08 2017

A134021 Length of n in balanced ternary representation.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Oct 19 2007

Shifted variant of A064099.

			100 = 1*3^4+1*3^3-1*3^2+0*3^1+1*3^0: a(100) = |++-0+| = 5.
200 = 1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0: a(200) = |+-+++-| = 6.
300 = 1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0: a(300) = |++-0+0| = 6.

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175.

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

Cf. A005812, A059095, A081604, A134022, A134023, A134024, A134025, A134026, A134027, A134028, A134421.

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

def a(n):
    if n==0: return 1
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

For n > 0: a(n) = ceiling(log(2*n+1)/log(3)).
a(n) = A134022(n) + A134023(n) + A134024(n).
0 <= a(n) - A081604(n) <= 1.
a(A134025(n)) = A081604(A134025(n)); a(A134026(n)) = A081604(A134026(n))+1.
a(A134027(n)) = a(n); a(abs(A134028(n))) <= a(n).
a(n) = A064099(n-1) for n>1.
n = Sum_{k=0..a(n)-1} (A059095(A134421(n)-2-k)*3^k), for n > 0. - Reinhard Zumkeller, Oct 25 2007
a(n) = A005812(n) + A134023(n).

A157671 Numbers whose ternary representation begins with 2.

Original entry on oeis.org

2, 6, 7, 8, 18, 19, 20, 21, 22, 23, 24, 25, 26, 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, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184

Offset: 1

Zak Seidov, Mar 04 2009

From R. J. Mathar, Mar 03 2009: (Start)
If we look at the sequence first differences, i.e.,
2, 4, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 82, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, we obtain the records in A034472. (End)
The lower and upper asymptotic densities of this sequence are 1/4 and 1/2, respectively. - Amiram Eldar, Feb 28 2021

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

Cf. A034472, A132141, A171960.

Subsequence of A134026. - Reinhard Zumkeller, Jan 20 2010

a157671 n = a157671_list !! (n-1)
a157671_list = filter ((== 2) . until (< 3) (flip div 3)) [1..]
-- Reinhard Zumkeller, Feb 06 2015

for n from 1 to 300 do dgs := convert(n,base,3) ; if op(-1,dgs) = 2 then printf("%d,",n) ; fi; od: # R. J. Mathar, Mar 03 2009

Flatten[(Range[2*3^#,3^(#+1)-1])&/@Range[0,4]]
Select[Range[200],First[IntegerDigits[#,3]]==2&] (* Harvey P. Dale, Oct 16 2012 *)
Table[FromDigits[#,3]&/@(Join[{2},#]&/@Tuples[{0,1,2},n]),{n,0,4}]// Flatten (* Harvey P. Dale, Jan 28 2022 *)

s=[];for(n=0,4,for(x=3^n,2*3^n-1,s=concat(s,x)));s

A number k is a term if and only if 2*3^m <= k <= 3^(m+1)-1, for m=0,1,2,...

A171960(a(n)) < a(n). - Reinhard Zumkeller, Jan 20 2010

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).

A134025 Numbers for which the balanced ternary representation is the same length as the ternary representation.

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 11, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121

Offset: 0

Reinhard Zumkeller, Oct 19 2007

A003462 is a subsequence; A171960(a(n)) >= a(n). - Reinhard Zumkeller, Jan 20 2010

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

Complement of A134026.

0,seq($3^(d-1)..floor(3^d/2), d=0..5); # Robert Israel, Dec 14 2015

f[n_, m_, k_] := If[n == 0, k, If[k < (3^(m + 1) - 1)/2, f[n - 1, m, k + 1], f[n - 1, m + 1, 3^(m + 1)]]]; Table[f[n, 0, 0], {n, 0, 63}] (* L. Edson Jeffery, Dec 10 2015 *)

a(n) = f(n,0,0) with f(n,m,k) = if n=0 then k else if k<(3^(m+1)-1)/2 then f(n-1,m,k+1) else f(n-1,m+1,3^(m+1)).
A134021(a(n)) = A081604(a(n)).
G.f.: x/(1-x)^2 + (1-x)^(-1)*Sum_{j>=1} ((3^j-1)/2) * x^(3/4 + 3^j/2 + j/2). - Robert Israel, Dec 14 2015

A216731 Primes p > 3 such that there is no power of 3 in the open interval (2p, 3p).

Original entry on oeis.org

5, 7, 17, 19, 23, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509

Offset: 1

N. J. A. Sloane, Sep 17 2012

nonn

Is this (apart from 2) the subset of primes in A134026? - R. J. Mathar, Sep 17 2012

Robert Israel, Table of n, a(n) for n = 1..10000
Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv preprint arXiv:1203.3969 [math.NT], 2012.

isA216731 := proc(n)
    if isprime(n) then
        floor(log[3](2*n)) = floor(log[3](3*n)) ;
    else
        false;
    end if;
end proc:
for n from 2 to 250 do
    p := ithprime(n) ;
    if isA216731(p) then
        printf("%d,",p) ;
    end if;
end do: # R. J. Mathar, Sep 17 2012

isA216731[n_] := If[PrimeQ[n], Floor[Log[3, 2*n]] == Floor[Log[3, 3*n]], False]; Reap[For[n = 2, n <= 100, n++, p = Prime[n]; If[isA216731[p], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Mar 06 2014, after R. J. Mathar *)

Name corrected by Robert Israel, May 11 2025

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A081604 Number of digits in ternary representation of n.

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A134021 Length of n in balanced ternary representation.

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A157671 Numbers whose ternary representation begins with 2.

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

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A134025 Numbers for which the balanced ternary representation is the same length as the ternary representation.

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A216731 Primes p > 3 such that there is no power of 3 in the open interval (2p, 3p).

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