A174658 -id:A174658 - cp's OEIS frontend

A257869 Nonnegative integers with an equal number of occurrences of all trits in balanced ternary representation.

6, 8, 136, 138, 144, 154, 156, 160, 164, 168, 170, 180, 186, 188, 208, 210, 214, 218, 222, 224, 232, 236, 248, 258, 260, 266, 288, 294, 296, 312, 314, 320, 3406, 3412, 3414, 3430, 3432, 3438, 3484, 3486, 3492, 3510, 3568, 3574, 3576, 3592, 3594, 3600, 3622

Offset: 1

Alois P. Heinz, May 11 2015

			6 = 1L0_bal3, 8 = 10L_bal3, 136 = 1LL001_bal3, 138 = 1LL010_bal3, 144 = 1LL100_bal3, where L represents (-1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Balanced ternary

Cf. A117967, A140267, A257867, A257868, A258410.

Subsequence of A174658.

p:= proc(n) local d, m, r; m:=n; r:=0;
      while m>0 do
        d:= irem(m, 3, 'm');
        if d=2 then m:=m+1 fi;
        r:= r+x^d
      od;
      simplify(r/(1+x+x^2))::integer
    end:
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1)) by 1
      while not p(k) do od; k
    end:
seq(a(n), n=1..70);

def a(n):
    s=[]
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        s.append(x)
    return s
print([n for n in range(1, 5001) if a(n).count(1)==a(n).count(-1) and a(n).count(-1)==a(n).count(0)]) # Indranil Ghosh, Jun 07 2017

A174657 Balanced ternary numbers with more negative trits than positive trits.

Original entry on oeis.org

5, 14, 15, 17, 23, 41, 42, 43, 44, 45, 47, 50, 51, 53, 59, 68, 69, 71, 77, 95, 122, 123, 124, 125, 126, 127, 128, 129, 131, 132, 133, 134, 135, 137, 140, 141, 143, 149, 150, 151, 152, 153, 155, 158, 159, 161, 167, 176, 177, 179, 185, 203, 204, 205, 206, 207, 209

Offset: 1

Daniel Forgues, Mar 26 2010

Numbers for which the sum of trits is negative.

Daniel Forgues, Table of n, a(n) for n=1..32685

Cf. A174658, A174659, A065363, A140267.

(* First run the program for A065363 to define balTernDigits *) Select[Range[210], Count[balTernDigits[#], -1] > Count[balTernDigits[#], 1] &] (* Alonso del Arte, Feb 26 2011 *)

def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        s+=x
    return s
print([n for n in range(301) if a(n)<0]) # Indranil Ghosh, Jun 07 2017

A174659 Numbers whose balanced ternary representation has more positive trits than negative trits.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 11, 12, 13, 19, 21, 22, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 49, 55, 57, 58, 61, 63, 64, 65, 66, 67, 73, 75, 76, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112

Offset: 1

Daniel Forgues, Mar 26 2010

Numbers for which the sum of trits is positive.

Daniel Forgues, Table of n, a(n) for n=1..53007

Cf. A174657, A174658, A065363, A140267.

(* First run the program for A065363 to define balTernDigits *) Select[Range[210], Count[balTernDigits[#], -1] < Count[balTernDigits[#], 1] &] (* Alonso del Arte, Feb 26 2011 *)

def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        s+=x
    return s
print([n for n in range(301) if a(n)>0]) # Indranil Ghosh, Jun 07 2017

a(n) < 2n. - Yifan Xie, Dec 24 2024

A350229 a(n) is the sum of n and the balanced ternary digits in n.

Original entry on oeis.org

0, 2, 2, 4, 6, 4, 6, 8, 8, 10, 12, 12, 14, 16, 12, 14, 16, 16, 18, 20, 20, 22, 24, 22, 24, 26, 26, 28, 30, 30, 32, 34, 32, 34, 36, 36, 38, 40, 40, 42, 44, 38, 40, 42, 42, 44, 46, 46, 48, 50, 48, 50, 52, 52, 54, 56, 56, 58, 60, 58, 60, 62, 62, 64, 66, 66, 68

Offset: 0

Rémy Sigrist, Jan 09 2022

The image of this sequence is the set of nonnegative even numbers (A005843).

			For n = 42:
- the balanced ternary representation of 42 is "1TTT0",
- so a(42) = 42 + 1 - 1 - 1 - 1 + 0 = 40.

See A062028, A092391, A230641 for similar sequences.

Cf. A005843, A065363, A174658 (fixed points).

Array[# + Total[If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[#, 3], 0] //. {x___, y_, k_ /; k > 1, z___} :> {x, y + 1, k - 3, z}]] &, 70, 0] (* Michael De Vlieger, Jan 15 2022 *)

a(n) = my (v=n, d); while (n, n=(n-d=[0,1,-1][1+n%3])/3; v+=d); v

a(n) = n + A065363(n).

a(n) = n iff n belongs to A174658.

A343604 a(n) is the least number > n with the same sum of balanced ternary digits as n.

Original entry on oeis.org

2, 3, 6, 7, 10, 15, 8, 9, 16, 11, 12, 19, 22, 31, 42, 17, 18, 23, 20, 21, 24, 25, 28, 43, 26, 27, 32, 29, 30, 33, 34, 37, 46, 35, 36, 49, 38, 39, 58, 67, 94, 123, 44, 45, 50, 47, 48, 51, 52, 55, 68, 53, 54, 59, 56, 57, 60, 61, 64, 69, 62, 63, 70, 65, 66, 73

Offset: 0

Rémy Sigrist, Apr 22 2021

This sequence can be extended to negative indexes by setting a(-n) = -A343605(n) for any n > 0.

			The first terms, in base 10 and in balanced ternary (where T denotes the digit -1), alongside A065363(n), are:
  n   a(n)  bter(n)  bter(a(n))  A065363(n)
  --  ----  -------  ----------  ----------
   0     2        0          1T           0
   1     3        1          10           1
   2     6       1T         1T0           0
   3     7       10         1T1           1
   4    10       11         101           2
   5    15      1TT        1TT0          -1
   6     8      1T0         10T           0
   7     9      1T1         100           1
   8    16      10T        1TT1           0
   9    11      100         11T           1
  10    12      101         110           2
  11    19      11T        1T01           1
  12    22      110        1T11           2

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A057168, A065363, A174658, A228915, A319021, A343605.

A065363(n) = { my (v=0, d); while (n, v+=d=centerlift(Mod(n,3)); n=(n-d)\3); v }
a(n) = my (s=A065363(n)); for (k=n+1, oo, if (s==A065363(k), return (k)))

a(9*n) = 9*n + 2.

a(A174658(n)) = A174658(n+1).

A343605 a(n) is the greatest number < n with the same sum of balanced ternary digits as n.

Original entry on oeis.org

-2, -5, 0, 1, -14, -1, 2, 3, 6, 7, 4, 9, 10, -41, -4, 5, 8, 15, 16, 11, 18, 19, 12, 17, 20, 21, 24, 25, 22, 27, 28, 13, 26, 29, 30, 33, 34, 31, 36, 37, -122, -13, 14, 23, 42, 43, 32, 45, 46, 35, 44, 47, 48, 51, 52, 49, 54, 55, 38, 53, 56, 57, 60, 61, 58, 63

Offset: 0

Rémy Sigrist, Apr 22 2021

This sequence can be extended to negative indexes by setting a(-n) = -A343604(n) for any n > 0.

			The first terms, in base 10 and in balanced ternary (where T denotes the digit -1), alongside A065363(n), are:
  n   a(n)  bter(n)  bter(a(n))  A065363(n)
  --  ----  -------  ----------  ----------
   0    -2        0          T1           0
   1    -5        1         T11           1
   2     0       1T           0           0
   3     1       10           1           1
   4   -14       11        T111           2
   5    -1      1TT           T          -1
   6     2      1T0          1T           0
   7     3      1T1          10           1
   8     6      10T         1T0           0
   9     7      100         1T1           1
  10     4      101          11           2
  11     9      11T         100           1
  12    10      110         101           2

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A003462, A007051, A065363, A174658, A343604.

A065363(n) = { my (v=0, d); while (n, v+=d=centerlift(Mod(n,3)); n=(n-d)\3); v }
a(n) = my (s=A065363(n)); forstep (k=n-1, -oo, -1, if (s==A065363(k), return (k)))

a(9*n) = 9*n - 2.
a(A174658(n+1)) = A174658(n) for any n > 0.
a(n) <= 0 iff n belongs to A003462 or to A007051.
a(A003462(k)) = -A007051(k + 1) for any k >= 0.
a(A007051(k)) = -A003462(k - 1) for any k > 0.

A355639 a(n) is the least k > 0 such that the balanced ternary expansion of k*n contains as many negative trits as positive trits.

Original entry on oeis.org

1, 2, 1, 2, 2, 4, 1, 8, 1, 2, 2, 14, 2, 2, 4, 4, 1, 8, 1, 14, 1, 8, 7, 2, 1, 16, 1, 2, 2, 8, 2, 2, 1, 14, 4, 2, 2, 2, 7, 2, 2, 4, 4, 2, 10, 4, 1, 4, 1, 2, 8, 8, 1, 8, 1, 8, 1, 14, 4, 4, 1, 8, 1, 8, 5, 2, 7, 14, 2, 2, 1, 2, 1, 2, 1, 16, 7, 2, 1, 8, 1, 2, 2, 8

Offset: 0

Rémy Sigrist, Jul 11 2022

The sequence is well defined: for n > 0, by the pigeonhole principle, there are necessarily two distinct integers i and j (say with i > j) such that 3^i == 3^j (mod n); the value 3^i - 3^j is a positive multiple of n containing exactly one positive trit and one negative trit, so a(n) <= (3^i - 3^j) / n.

			For n = 5:
- the first multiple of 5 (alongside their balanced ternary expansions) are:
      k  k*5  bter(k*5)  #1  #T
      -  ---  ---------  --  --
      1    5        1TT   1   2
      2   10        101   2   0
      3   15       1TT0   1   2
      4   20       1T1T   2   2
- negative and positive trits are first balanced for k = 4,
- so a(5) = 4.

See A351599 for a similar sequence.

Cf. A065363, A174658, A355640.

a(n) = { for (k=1, oo, my (m=k*n, s=0, d); while (m, m=(m-d=[0,1,-1][1+m%3])/3; s+=d); if (s==0, return (k))) }

a(n) = 1 iff n belongs to A174658.

A283800 Numbers such that the sum of trits of its balanced ternary representation is 1 or -1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 33, 35, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 95, 97, 99, 101, 105, 107, 113, 127, 129, 133, 135, 137, 139, 141, 143, 145, 147, 151, 153, 155, 157, 159, 161

Offset: 1

Lei Zhou, Mar 16 2017

			3 = 10 in balanced ternary (bt) notation, 1+0 = 1, so 3 is in the list;
...
11 = 11T in bt notation, 1+1+T = 1, here T represent -1, so 11 is in the list;
13 = 111 in bt notation, 1+1+1 = 3, so 13 is NOT in the list.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A065303, A174658.

BTDigits[m_Integer,
   g_] :=(*This is to determine digits of a number in balanced \
ternary notation.*)
  Module[{n = m, d, sign, t = g},
   If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];
    d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];
    While[Length[t] < d, PrependTo[t, 0]];
    t[[Length[t] + 1 - d]] = sign;
    t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
n = 0; Table[While[n++; g = {}; bt = BTDigits[n, g]; s = Total[bt];
  Abs[s] != 1]; n, {i, 1, 61}]

A355640 a(0) = 0, and for any n > 0, a(n) is the least positive multiple of n whose balanced ternary expansion contains as many negative trits as positive trits.

Original entry on oeis.org

0, 2, 2, 6, 8, 20, 6, 56, 8, 18, 20, 154, 24, 26, 56, 60, 16, 136, 18, 266, 20, 168, 154, 46, 24, 400, 26, 54, 56, 232, 60, 62, 32, 462, 136, 70, 72, 74, 266, 78, 80, 164, 168, 86, 440, 180, 46, 188, 48, 98, 400, 408, 52, 424, 54, 440, 56, 798, 232, 236, 60

Offset: 0

Rémy Sigrist, Jul 11 2022

A174658 corresponds to fixed points.

			For n = 5:
- the first multiple of 5 (alongside their balanced ternary expansions) are:
      k  k*5  bter(k*5)  #1  #T
      -  ---  ---------  --  --
      1    5        1TT   1   2
      2   10        101   2   0
      3   15       1TT0   1   2
      4   20       1T1T   2   2
- negative and positive trits are first balanced for k = 4,
- so a(5) = 4*5 = 20.

See A143146 for a similar sequence.

Cf. A065363, A174658 (fixed points), A355639.

a(n) = { for (k=1, oo, my (m=k*n, s=0, d); while (m, m=(m-d=[0, 1, -1][1+m%3])/3; s+=d); if (s==0, return (k*n))) }

a(n) = n * A355639(n).

A355642 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the balanced ternary expansion of n * a(n) contains as many negative trits as positive trits.

Original entry on oeis.org

0, 2, 1, 6, 4, 12, 3, 8, 7, 16, 13, 14, 5, 10, 11, 26, 9, 24, 23, 28, 20, 22, 21, 18, 17, 32, 15, 46, 19, 30, 29, 40, 25, 42, 34, 48, 36, 38, 37, 44, 31, 52, 33, 54, 39, 58, 27, 74, 35, 56, 55, 68, 41, 70, 43, 50, 49, 64, 45, 66, 60, 72, 63, 62, 57, 76, 59, 80

Offset: 0

Rémy Sigrist, Jul 11 2022

This sequence is a self-inverse permutation of the nonnegative integers.

			The first terms, alongside the balanced ternary expansion of n * a(n), are:
  n   a(n)  bter(n*a(n))
  --  ----  ------------
   0     0             0
   1     2            1T
   2     1            1T
   3     6          1T00
   4     4          1TT1
   5    12         1T1T0
   6     3          1T00
   7     8         1T01T
   8     7         1T01T
   9    16        1TT100
  10    13        1TTT11
  11    14        1T0T01
  12     5         1T1T0

Rémy Sigrist, Table of n, a(n) for n = 0..2187
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A306993 for a similar sequence.

Cf. A065363, A174658, A355639.

PARI
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A257869 Nonnegative integers with an equal number of occurrences of all trits in balanced ternary representation.

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A174657 Balanced ternary numbers with more negative trits than positive trits.

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A174659 Numbers whose balanced ternary representation has more positive trits than negative trits.

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A350229 a(n) is the sum of n and the balanced ternary digits in n.

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A343604 a(n) is the least number > n with the same sum of balanced ternary digits as n.

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A343605 a(n) is the greatest number < n with the same sum of balanced ternary digits as n.

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A355639 a(n) is the least k > 0 such that the balanced ternary expansion of k*n contains as many negative trits as positive trits.

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A283800 Numbers such that the sum of trits of its balanced ternary representation is 1 or -1.

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