A319021 -id:A319021 - cp's OEIS frontend

A319023 Let S be the sequence generated by these rules: 1 is in S, and if k is in S, then A057168(k) and A319021(k) are in S, and duplicates are deleted as they occur; a(n) is the n-th term of S.

1, 2, 3, 4, 5, 9, 8, 6, 7, 10, 27, 16, 14, 11, 12, 29, 81, 32, 20, 19, 13, 17, 18, 30, 31, 82, 243, 64, 34, 24, 22, 21, 15, 23, 28, 39, 36, 47, 33, 84, 245, 729, 128, 66, 38, 25, 35, 43, 45, 40, 54, 55, 49, 37, 88, 90, 246, 247, 730, 2187, 256, 130, 68, 72, 41

Offset: 1

Rémy Sigrist, Sep 08 2018

This sequence is a permutation of the natural numbers (with inverse A319024):
- this sequence is injective,
- this sequence is surjective: by contradiction:
- let m be the least integer missing from the sequence,
- as a(1) = 1, we have m > 1,
- also, m belongs to A000225 and to A062318,
- however the only positive integer belonging to both sequences is 1,
- hence a contradiction, QED.

			The first terms, alongside b(n) = A057168(a(n)) and t(n) = A319021(a(n)), are:
  n   a(n)  b(n)  t(n)
  --  ----  ----  ----
   1     1     2     3
   2     2     4     4
   3     3     5     9
   4     4     8     6
   5     5     6     7
   6     9    10    27
   7     8    16    14
   8     6     9    10
   9     7    11    11
  10    10    12    12
  11    27    29    81
  12    16    32    20
  13    14    19    16
  14    11    13    13
  15    12    17    18
  16    29    30    31
  17    81    82   243
  18    32    64    34
  19    20    24    22
  20    19    21    21

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A319023
Index entries for sequences that are permutations of the natural numbers

Cf. A000225, A057168, A062318, A319021, A319024 (inverse).

PARI
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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).

cp's OEIS Frontend

A319023 Let S be the sequence generated by these rules: 1 is in S, and if k is in S, then A057168(k) and A319021(k) are in S, and duplicates are deleted as they occur; a(n) is the n-th term of S.

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