A321682 Numbers with distinct digits in factorial base.

0, 1, 2, 4, 5, 10, 13, 14, 19, 20, 22, 23, 46, 67, 68, 77, 82, 85, 86, 101, 106, 109, 110, 115, 116, 118, 119, 238, 355, 356, 461, 466, 469, 470, 503, 526, 547, 548, 557, 562, 565, 566, 623, 646, 667, 668, 677, 682, 685, 686, 701, 706, 709, 710, 715, 716, 718

Offset: 1

Author

Rémy Sigrist, Nov 16 2018

Keywords

• nonn
• base

Comments

This sequence is a variant of A010784; however here we have infinitely many terms (for example all the terms of A033312 belong to this sequence).

Examples

			The first terms, alongside the corresponding factorial base representations, are:
  n   a(n)  fac(a(n))
  --  ----  ---------
   1     0        (0)
   2     1        (1)
   3     2      (1,0)
   4     4      (2,0)
   5     5      (2,1)
   6    10    (1,2,0)
   7    13    (2,0,1)
   8    14    (2,1,0)
   9    19    (3,0,1)
  10    20    (3,1,0)
  11    22    (3,2,0)
  12    23    (3,2,1)
  13    46  (1,3,2,0)
  14    67  (2,3,0,1)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8179 (terms up to 13!)
• Index entries for sequences related to factorial base representation.

Crossrefs

Cf. A010784, A033312, A108731.

Programs

Maple

b:= proc(n, i) local r; `if`(n (l-> is(nops(l)=nops({l[]})))(b(n, 2)):
select(t, [$0..1000])[];  # Alois P. Heinz, Nov 16 2018

Mathematica

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; UnsameQ @@ s]; Select[Range[0, 720], q] (* Amiram Eldar, Feb 21 2024 *)

PARI

is(n) = my (s=0); for (k=2, oo, if (n==0, return (1)); my (d=n%k); if (bittest(s,d), return (0), s+=2^d; n\=k))