A389644 Nonnegative numbers k such that for some integer m the factorial base expansion of k (without leading zeros), when read backwards, corresponds to that of m, and conversely, the factorial base expansion of m (without leading zeros), when read backwards, corresponds to that of k.

0, 1, 3, 7, 9, 11, 25, 27, 29, 31, 33, 35, 37, 39, 41, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 721, 723, 725, 727, 729

Offset

1,3

Comments

This sequence naturally contains the palindromes in factorial base (A046807).

A positive number k belongs to this sequence iff its factorial base expansion, say (d_1, ..., d_w), satisfies d_1 = d_w = 1 and d_i <= i for i in 1..w.

There are A010551(w+2)/4 terms with w digits in factorial base for w > 1.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..4515

Rémy Sigrist, PARI program

Index entries for sequences related to factorial base representation

Example

The first terms, alongside their factorial base expansion, are:
  n   a(n)  fact(a(n))
  --  ----  -----------
   1     0  ()
   2     1  (1)
   3     3  (1,1)
   4     7  (1,0,1)
   5     9  (1,1,1)
   6    11  (1,2,1)
   7    25  (1,0,0,1)
   8    27  (1,0,1,1)
   9    29  (1,0,2,1)
  10    31  (1,1,0,1)
  11    33  (1,1,1,1)
  12    35  (1,1,2,1)
  13    37  (1,2,0,1)
  14    39  (1,2,1,1)
  15    41  (1,2,2,1)
  16   121  (1,0,0,0,1)
  17   123  (1,0,0,1,1)
  18   125  (1,0,0,2,1)

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A010551, A108731, A046807, A389645.

Sequence in context: A336655 A003538 A018596 * A191181 A046807 A333423

Adjacent sequences: A389641 A389642 A389643 * A389645 A389646 A389647

Keyword

nonn,base

Author

Rémy Sigrist, Oct 09 2025

Status

approved