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A382184 a(n) is the least k >= 0 such that the factorial base expansion of n starts with that of k while the remaining digits are zeros.

%I A382184 #9 Mar 20 2025 09:29:22
%S A382184 0,1,1,3,4,5,1,7,3,9,10,11,4,13,5,15,16,17,18,19,20,21,22,23,1,25,7,
%T A382184 27,28,29,3,31,9,33,34,35,10,37,11,39,40,41,42,43,44,45,46,47,4,49,13,
%U A382184 51,52,53,5,55,15,57,58,59,16,61,17,63,64,65,66,67,68,69
%N A382184 a(n) is the least k >= 0 such that the factorial base expansion of n starts with that of k while the remaining digits are zeros.
%H A382184 Rémy Sigrist, <a href="/A382184/b382184.txt">Table of n, a(n) for n = 0..10000</a>
%H A382184 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A382184 a(n) <= n with equality iff n = 0 or n belongs to A273670.
%F A382184 a(k!) = 1 for any k >= 0.
%e A382184 The first terms, in decimal and in factorial base, are:
%e A382184   n   a(n)  fact(n)  fact(a(n))
%e A382184   --  ----  -------  ----------
%e A382184    0     0  0        0
%e A382184    1     1  1        1
%e A382184    2     1  1,0      1
%e A382184    3     3  1,1      1,1
%e A382184    4     4  2,0      2,0
%e A382184    5     5  2,1      2,1
%e A382184    6     1  1,0,0    1
%e A382184    7     7  1,0,1    1,0,1
%e A382184    8     3  1,1,0    1,1
%e A382184    9     9  1,1,1    1,1,1
%e A382184   10    10  1,2,0    1,2,0
%e A382184   11    11  1,2,1    1,2,1
%e A382184   12     4  2,0,0    2,0
%e A382184   13    13  2,0,1    2,0,1
%e A382184   14     5  2,1,0    2,1
%e A382184   15    15  2,1,1    2,1,1
%o A382184 (PARI) a(n) = { if (n, my (m = n, s = oo, d); for (r = 2, oo, if (m==0 || s==0, break, d = m%r, s = min(s, r-1-d);); m \= r;); if (s, my (v = 0); for (r = 2, oo, if (n==0, return (v), v += (n%r) * max(0, r-1-s)!; n \= r;);););); return (n); }
%Y A382184 Cf. A000265, A004151, A153880, A273670, A382177, A382217.
%K A382184 nonn,base
%O A382184 0,4
%A A382184 _Rémy Sigrist_, Mar 17 2025