A382269 -id:A382269 - cp's OEIS frontend

A382263 a(n) is the unique k such that the factorial base expansion of A382262(n) is, when read from right to left, the ordinal transform of that of A382262(k).

0, 1, 3, 2, 7, 6, 5, 4, 17, 16, 15, 12, 11, 14, 13, 10, 9, 8, 43, 42, 41, 37, 40, 39, 38, 36, 35, 28, 27, 34, 33, 32, 31, 30, 29, 26, 25, 21, 24, 23, 22, 20, 19, 18, 119, 118, 117, 112, 116, 114, 113, 111, 110, 95, 115, 109, 108, 99, 107, 97, 96, 106, 105, 104

Offset: 0

Rémy Sigrist, Mar 19 2025

A self-inverse permutation of the nonnegative integers with only two fixed points: a(0) = 0 and a(1) = 1.

			The first terms, alongside the corresponding factorial base expansions, are:
  n   a(n)  fact(A382262(n))  fact(A382262(a(n)))
  --  ----  ----------------  -------------------
   0     0  0                 0
   1     1  1                 1
   2     3  1,1               2,1
   3     2  2,1               1,1
   4     7  1,1,1             3,2,1
   5     6  1,2,1             2,1,1
   6     5  2,1,1             1,2,1
   7     4  3,2,1             1,1,1
   8    17  1,1,1,1           4,3,2,1
   9    16  1,1,2,1           3,2,1,1
  10    15  1,2,1,1           3,1,2,1
  11    12  1,3,2,1           2,1,1,1
  12    11  2,1,1,1           1,3,2,1
  13    14  2,1,2,1           2,2,1,1
  14    13  2,2,1,1           2,1,2,1
  15    10  3,1,2,1           1,2,1,1
  16     9  3,2,1,1           1,1,2,1
  17     8  4,3,2,1           1,1,1,1

Rémy Sigrist, Table of n, a(n) for n = 0..13231
Rémy Sigrist, PARI program
Index entries for sequences related to factorial base representation
Index entries for sequences that are permutations of the natural numbers

Cf. A382262, A382269.

PARI
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See Links section.
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A382269(A382262(a(n))) = A382262(n).

A382313 The factorial base expansion of a(n) corresponds to the restricted growth sequence of that of n (when read from right to left).

Original entry on oeis.org

0, 1, 5, 3, 5, 5, 15, 11, 17, 9, 23, 11, 15, 23, 23, 15, 17, 17, 15, 23, 23, 15, 23, 23, 57, 41, 59, 39, 83, 47, 63, 35, 65, 33, 95, 35, 87, 47, 71, 39, 89, 41, 87, 47, 71, 39, 119, 47, 57, 89, 83, 87, 59, 71, 87, 83, 89, 57, 71, 59, 63, 95, 95, 63, 65, 65, 87

Offset: 0

Rémy Sigrist, Mar 21 2025

We ignore nonleading zeros in factorial base expansions.

All terms belong to A120696.

			The first terms, in decimal and in factorial base, are:
  n   a(n)  fact(n)  fact(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     1        1           1
   2     5       10          21
   3     3       11          11
   4     5       20          21
   5     5       21          21
   6    15      100         211
   7    11      101         121
   8    17      110         221
   9     9      111         111
  10    23      120         321
  11    11      121         121
  12    15      200         211
  13    23      201         321
  14    23      210         321
  15    15      211         211

See A382269 for a similar sequence.

Cf. A120696.

a(n) = { my (v = 0, m = Map(), u = 0, d, c); for (r = 2, oo, if (n==0, return (v), d = n%r; n \= r; if (!mapisdefined(m, d, &c), mapput(m, d, c=u++);); v += c*(r-1)!;);); }

a(a(n)) = a(n).

cp's OEIS Frontend

A382263 a(n) is the unique k such that the factorial base expansion of A382262(n) is, when read from right to left, the ordinal transform of that of A382262(k).

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