A157671 -id:A157671 - cp's OEIS frontend

A132141 Numbers whose ternary representation begins with 1.

1, 3, 4, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 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, 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

Offset: 1

Reinhard Zumkeller, Aug 20 2007

The lower and upper asymptotic densities of this sequence are 1/2 and 3/4, respectively. - Amiram Eldar, Feb 28 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz and Michael Someck, Four quotient set gems, The American Mathematical Monthly, Vol. 121, No. 7 (2014), pp. 590-598; arXiv preprint, arXiv:1312.1036 [math.NT], 2013.
Christian Mauduit, Propriétés arithmétiques des substitutions, in Séminaire de Théorie des Nombres, Paris, 1989-90, pp. 177-190 (in French).
Index entries for 3-automatic sequences.

Cf. A007089, A132140, A157671, A131835.

a132141 n = a132141_list !! (n-1)
a132141_list = filter ((== 1) . until (< 3) (flip div 3)) [1..]
-- Reinhard Zumkeller, Feb 06 2015

Flatten[(Range[3^#,2 3^#-1])&/@Range[0,4]] (* Zak Seidov, Mar 03 2009 *)

s=[];for(n=0,4,for(x=3^n,2*3^n-1,s=concat(s,x)));s \\ Zak Seidov, Mar 03 2009

a(n) = n + 3^logint(n<<1,3) >> 1; \\ Kevin Ryde, Feb 19 2022

A number n is a term iff 3^m <= n < 2*3^m -1, for m=0,1,2,... - Zak Seidov, Mar 03 2009

a(n) = n + (3^floor(log_3(2*n)) - 1)/2. - Kevin Ryde, Feb 19 2022

A134026 Numbers that are in balanced ternary representation longer than in ternary representation.

Original entry on oeis.org

2, 5, 6, 7, 8, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 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, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131

Offset: 1

Reinhard Zumkeller, Oct 19 2007

A157671 is a subsequence. - Reinhard Zumkeller, Jan 20 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A134025.

Cf. A081604, A134021, A157671, A171960.

Select[Range[150], Ceiling[Log[3, 2*#+1]] > IntegerLength[#, 3] &] (* Amiram Eldar, Apr 03 2025 *)

a(n) = g(n,0,0) with g(n,m,k) = if n=0 then k else if k=3^m-1 then g(n-1,m,3*(k+1)/2+1) else g(n-1,m,k+1).
A134021(a(n)) = A081604(a(n)) + 1.
A171960(a(n)) < a(n). - Reinhard Zumkeller, Jan 20 2010

A338246 Nonpositive values in A117966, in order of appearance and negated.

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 8, 10, 6, 5, 7, 12, 11, 13, 27, 26, 28, 24, 23, 25, 30, 29, 31, 18, 17, 19, 15, 14, 16, 21, 20, 22, 36, 35, 37, 33, 32, 34, 39, 38, 40, 81, 80, 82, 78, 77, 79, 84, 83, 85, 72, 71, 73, 69, 68, 70, 75, 74, 76, 90, 89, 91, 87, 86, 88, 93, 92, 94

Offset: 0

Rémy Sigrist, Oct 18 2020

This sequence is a self-inverse permutation of the nonnegative integers (the offset has been set to 0 so as to get a permutation).

			A117966 = 0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, -9, ...
We keep:  0,     1,           3,  2,  4,                                 9, ...

Cf. A003462 (fixed points), A117966, A157671, A338245.

A117966(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n, 3)), 'x), 'x, 3)
print (-select(v -> v<=0, apply(A117966, [0..188])))

a(0) = 0.
a(n) = -A117966(A157671(n)) for any n > 0.
a(n) = n iff n belongs to A003462.

A351702 In the balanced ternary representation of n, reverse the order of digits other than the most significant.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 11, 6, 9, 12, 7, 10, 13, 14, 23, 32, 17, 26, 35, 20, 29, 38, 15, 24, 33, 18, 27, 36, 21, 30, 39, 16, 25, 34, 19, 28, 37, 22, 31, 40, 41, 68, 95, 50, 77, 104, 59, 86, 113, 44, 71, 98, 53, 80, 107, 62, 89, 116, 47, 74, 101, 56, 83, 110, 65, 92

Offset: 0

Kevin Ryde, Feb 19 2022

Self-inverse permutation with swaps confined to terms of a given digit length (A134021) so within blocks n = (3^k+1)/2 .. (3^(k+1)-1)/2.
Can extend to negative n by a(-n) = -a(n).
A072998 is balanced ternary coded in decimal digits so that reversal except first digit of A072998(n) is at A072998(a(n)).  Similarly its ternary equivalent A157671, and also A132141 ternary starting with 1.
These sequences all have a fixed initial digit followed by all ternary strings which is the reversed part.  A007932 is such strings as decimal digits 1,2,3 but it omits the empty string so the whole reversal of A007932(n) is at A007932(a(n+1)-1).
Fixed points a(n) = n are where n in balanced ternary is a palindrome apart from its initial 1.  These are the full balanced ternary palindromes with their least significant 1 removed, so all n = (A134027(m)-1)/3 for m>=2.

			n    = 224 = balanced ternary 1,  0, -1, 1,  0, -1
                      reverse     ^^^^^^^^^^^^^^^^
a(n) = 168 = balanced ternary 1, -1,  0, 1, -1,  0

Cf. A059095 (balanced ternary), A134028 (full reverse), A134027 (palindromes).
Cf. A072998, A157671, A132141, A007932.
In other bases: A059893 (binary), A343150 (Zeckendorf), A343152 (lazy Fibonacci).

a(n) = if(n==0,0, my(k=if(n,logint(n<<1,3)), s=(3^k+1)>>1); s + fromdigits(Vec(Vecrev(digits(n-s,3)),k),3));

A328749 a(n) = Sum_{k = 0..w and t_k > 0} (-1)^t_k * 2^k, where Sum_{k = 0..w} t_k * 3^k is the ternary representation of n.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Oct 27 2019

Every integer appears in the sequence.

			a(42) = a(1*3^3 + 1*3^2 + 2*3^1) = -2^3 - 2^2 + 2^1 = -10.

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A004488, A132141, A157671, A328728.

a(n) = my (d=Vecrev(digits(n,3))); sum(i=1, #d, if (d[i], (2^i) * (-1)^d[i], 0))/2

from sympy.ntheory.factor_ import digits
def A328749(n): return sum((-(1<0) # Chai Wah Wu, Apr 12 2023

a(n) = 0 iff n = 0.
a(n) > 0 iff n belongs to A157671.
a(n) < 0 iff n belongs to A132141.
a(A004488(n)) = -a(n).

A333773 Replace 2's with (-1)'s in ternary representation of n and sum nonzero terms with alternating signs.

Original entry on oeis.org

0, 1, -1, 3, 2, 4, -3, -4, -2, 9, 8, 10, 6, 7, 5, 12, 13, 11, -9, -10, -8, -12, -11, -13, -6, -5, -7, 27, 26, 28, 24, 25, 23, 30, 31, 29, 18, 19, 17, 21, 20, 22, 15, 14, 16, 36, 37, 35, 39, 38, 40, 33, 32, 34, -27, -28, -26, -30, -29, -31, -24, -23, -25, -36

Offset: 0

Rémy Sigrist, Apr 05 2020

This sequence is a variant of A117966, and shares features with A065620.

Every integer appears exactly once in this sequence.

			For n = 97:
- 97 = 3^4 + 3^2 + 2*3^1 + 3^0,
- hence a(97) = 3^4 - 3^2 + (-1)*3^1 - 3^0 = 68.

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A004488, A024023, A065620, A117966, A132141, A157671, A160384, A333780.

a(n) = { my (v=0, t=Vecrev(digits(n,3))); for (k=1, #t, if (t[k]==1, v=+3^(k-1)-v, t[k]==2, v=-3^(k-1)-v)); v }

a(3*n)   = 3*a(n).
a(3*n+1) = 3*a(n) + (-1)^A160384(n).
a(3*n+2) = 3*a(n) - (-1)^A160384(n).
Sum_{k=0..n} a(k) >= 0 with equality iff n belongs to A024023.
a(n) > 0 iff n belongs to A132141.
a(n) < 0 iff n belongs to A157671.
a(A004488(n)) = -a(n).

A370925 Rectangular array, read by antidiagonals: row n consists of the numbers m whose ternary representation starts with 2 and has exactly n runs.

Original entry on oeis.org

2, 8, 6, 26, 7, 19, 80, 18, 20, 57, 242, 22, 21, 59, 172, 728, 24, 23, 60, 173, 516, 2186, 25, 55, 61, 177, 518, 1549, 6560, 54, 56, 64, 178, 519, 1550, 4647, 19682, 67, 58, 65, 181, 520, 1554, 4649, 13942

Offset: 1

Clark Kimberling, Mar 13 2024

Every positive integer occurs in this array or A370924.

			Corner:
       2     8    26    80   242   728  2186
       6     7    18    22    24    25    54
      19    20    21    23    55    56    58
      57    59    60    61    64    65    69
     172   173   177   178   181   182   183
     516   518   519   520   532   533   534
    1549  1550  1554  1555  1558  1559  1560

Cf. A007089, A157671, A370893, A370924.

d[n_] := First[IntegerDigits[n, 3]];
a[n_] := a[n] = Select[Range[30000],
d[#] == 2 && Length[Split[IntegerDigits[#, 3]]] == n &];
t[n_, k_] := a[n][[k]];
Grid[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]] (* array *)
Table[t[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* sequence *)

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A132141 Numbers whose ternary representation begins with 1.

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A134026 Numbers that are in balanced ternary representation longer than in ternary representation.

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A338246 Nonpositive values in A117966, in order of appearance and negated.

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A351702 In the balanced ternary representation of n, reverse the order of digits other than the most significant.

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A328749 a(n) = Sum_{k = 0..w and t_k > 0} (-1)^t_k * 2^k, where Sum_{k = 0..w} t_k * 3^k is the ternary representation of n.

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A333773 Replace 2's with (-1)'s in ternary representation of n and sum nonzero terms with alternating signs.

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A370925 Rectangular array, read by antidiagonals: row n consists of the numbers m whose ternary representation starts with 2 and has exactly n runs.

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