A389644 - OEIS

%I #9 Oct 16 2025 21:00:47

%S 0,1,3,7,9,11,25,27,29,31,33,35,37,39,41,121,123,125,127,129,131,133,

%T 135,137,139,141,143,145,147,149,151,153,155,157,159,161,163,165,167,

%U 169,171,173,175,177,179,181,183,185,187,189,191,721,723,725,727,729

%N 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.

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

%C 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.

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

%H Rémy Sigrist, <a href="/A389644/b389644.txt">Table of n, a(n) for n = 1..4515</a>

%H Rémy Sigrist, <a href="/A389644/a389644.gp.txt">PARI program</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%e The first terms, alongside their factorial base expansion, are:

%e   n   a(n)  fact(a(n))

%e   --  ----  -----------

%e    1     0  ()

%e    2     1  (1)

%e    3     3  (1,1)

%e    4     7  (1,0,1)

%e    5     9  (1,1,1)

%e    6    11  (1,2,1)

%e    7    25  (1,0,0,1)

%e    8    27  (1,0,1,1)

%e    9    29  (1,0,2,1)

%e   10    31  (1,1,0,1)

%e   11    33  (1,1,1,1)

%e   12    35  (1,1,2,1)

%e   13    37  (1,2,0,1)

%e   14    39  (1,2,1,1)

%e   15    41  (1,2,2,1)

%e   16   121  (1,0,0,0,1)

%e   17   123  (1,0,0,1,1)

%e   18   125  (1,0,0,2,1)

%o (PARI) \\ See Links section.

%Y Cf. A010551, A108731, A046807, A389645.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Oct 09 2025