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A321682 Numbers with distinct digits in factorial base.

%I A321682 #21 Feb 21 2024 01:20:48
%S A321682 0,1,2,4,5,10,13,14,19,20,22,23,46,67,68,77,82,85,86,101,106,109,110,
%T A321682 115,116,118,119,238,355,356,461,466,469,470,503,526,547,548,557,562,
%U A321682 565,566,623,646,667,668,677,682,685,686,701,706,709,710,715,716,718
%N A321682 Numbers with distinct digits in factorial base.
%C A321682 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).
%H A321682 Rémy Sigrist, <a href="/A321682/b321682.txt">Table of n, a(n) for n = 1..8179</a> (terms up to 13!)
%H A321682 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%e A321682 The first terms, alongside the corresponding factorial base representations, are:
%e A321682   n   a(n)  fac(a(n))
%e A321682   --  ----  ---------
%e A321682    1     0        (0)
%e A321682    2     1        (1)
%e A321682    3     2      (1,0)
%e A321682    4     4      (2,0)
%e A321682    5     5      (2,1)
%e A321682    6    10    (1,2,0)
%e A321682    7    13    (2,0,1)
%e A321682    8    14    (2,1,0)
%e A321682    9    19    (3,0,1)
%e A321682   10    20    (3,1,0)
%e A321682   11    22    (3,2,0)
%e A321682   12    23    (3,2,1)
%e A321682   13    46  (1,3,2,0)
%e A321682   14    67  (2,3,0,1)
%p A321682 b:= proc(n, i) local r; `if`(n<i, [n],
%p A321682       [b(iquo(n, i, 'r'), i+1)[], r])
%p A321682     end:
%p A321682 t:= n-> (l-> is(nops(l)=nops({l[]})))(b(n, 2)):
%p A321682 select(t, [$0..1000])[];  # _Alois P. Heinz_, Nov 16 2018
%t A321682 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 *)
%o A321682 (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))
%Y A321682 Cf. A010784, A033312, A108731.
%K A321682 nonn,base
%O A321682 1,3
%A A321682 _Rémy Sigrist_, Nov 16 2018