A353830 The positions of nonzero digits in the balanced ternary expansions of n and a(n) are the same, and the k-th rightmost nonzero digit in a(n) equals the product of the k rightmost nonzero digits in n.

0, 1, -4, 3, 4, 11, -12, -11, -10, 9, 10, -13, 12, 13, -34, 33, 34, 35, -36, -35, 32, -33, -32, 29, -30, -29, -28, 27, 28, -31, 30, 31, 38, -39, -38, -37, 36, 37, -40, 39, 40, 101, -102, -101, -100, 99, 100, -103, 102, 103, -106, 105, 106, 107, -108, -107, 104

Offset: 0

Author

Rémy Sigrist, May 08 2022

Keywords

• sign
• base

Comments

This sequence can naturally be extended to negative integers; we then obtain a permutation of the integers (Z).

A number is a fixed point of this sequence iff it has no digit -1 in its balanced ternary expansion (A005836).

Examples

			The first terms, in decimal and in balanced ternary, are:
  n   a(n)  bter(n)  bter(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     1        1           1
   2    -4       1T          TT
   3     3       10          10
   4     4       11          11
   5    11      1TT         11T
   6   -12      1T0         TT0
   7   -11      1T1         TT1
   8   -10      10T         T0T
   9     9      100         100
  10    10      101         101
  11   -13      11T         TTT
  12    12      110         110
		

Links

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

Crossrefs

See A305458, A353824, A353826, A353828 for similar sequences.

Cf. A005836 (fixed points), A029858, A153775.

Programs

PARI

a(n) = {
        my (d=[], t, p=1);
        while (n, d=concat(t=[0,1,-1][1+n%3], d); n=(n-t)/3);
        forstep (k=#d, 1, -1, if (d[k], d[k]=p*=d[k]));
        fromdigits(d,3);
}

Formula

a(3*n) = 3*a(n).

a(3*n + 1) = 3*a(n) + 1.

Sum_{k = 0..n} a(n) = 0 iff n belongs to A029858.