A382262 -id:A382262 - 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).

A382269 The factorial base expansion of a(n) is, when read from right to left, the ordinal transform of that of n.

Original entry on oeis.org

0, 1, 3, 5, 3, 3, 11, 15, 15, 23, 9, 15, 11, 9, 9, 11, 15, 15, 11, 9, 9, 11, 9, 9, 47, 63, 63, 83, 39, 57, 59, 87, 87, 119, 57, 87, 35, 57, 57, 83, 39, 63, 35, 57, 57, 83, 33, 57, 47, 39, 39, 35, 63, 57, 35, 39, 39, 47, 57, 63, 59, 57, 57, 59, 87, 87, 35, 33

Offset: 0

Rémy Sigrist, Mar 20 2025

We ignore nonleading zeros in factorial base expansions.

All terms belong to A382262.

			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     3  1,0      1,1
   3     5  1,1      2,1
   4     3  2,0      1,1
   5     3  2,1      1,1
   6    11  1,0,0    1,2,1
   7    15  1,0,1    2,1,1
   8    15  1,1,0    2,1,1
   9    23  1,1,1    3,2,1
  10     9  1,2,0    1,1,1
  11    15  1,2,1    2,1,1
  12    11  2,0,0    1,2,1
  13     9  2,0,1    1,1,1
  14     9  2,1,0    1,1,1
  15    11  2,1,1    1,2,1
  16    15  2,2,0    2,1,1

Cf. A382262.

a(n) = { my (v = 0, d, f = [0]); for (r = 2, oo, if (n==0, return (v); ); if (#f < r, f = concat(f, vector(#f)); ); v += f[1+(n % r)]++*(r-1)!; n \= r; ); }

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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