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.

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..83.

Formula

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

Example

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.

Prog

(PARI) 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))) }

Crossrefs

See A351599 for a similar sequence.

Cf. A065363, A174658, A355640.

Sequence in context: A158379 A122838 A070306 * A050378 A225216 A161833

Adjacent sequences: A355636 A355637 A355638 * A355640 A355641 A355642

Keyword

nonn,base

Author

Rémy Sigrist, Jul 11 2022

Status

approved