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A382178 a(n) is the least k > 1 such that the factorial base expansion of k*n starts with that of n.

%I A382178 #8 Mar 20 2025 09:29:28
%S A382178 2,2,3,3,3,3,4,18,18,17,17,16,4,19,19,18,18,101,4,115,114,110,110,18,
%T A382178 5,203,199,192,189,183,28,187,27,179,177,1341,180,176,26,170,168,165,
%U A382178 1320,168,1277,1251,162,159,5,1649,204,1598,1579,1551,200,197,195
%N A382178 a(n) is the least k > 1 such that the factorial base expansion of k*n starts with that of n.
%C A382178 This sequence is well defined (see A382177).
%H A382178 Rémy Sigrist, <a href="/A382178/b382178.txt">Table of n, a(n) for n = 0..10000</a>
%H A382178 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A382178 a(n) <= A382177(n).
%e A382178 The first terms, in decimal and in factorial base, are:
%e A382178   n   a(n)  fact(n)  fact(a(n)*n)
%e A382178   --  ----  -------  ------------
%e A382178    0     2  0        0
%e A382178    1     2  1        1,0
%e A382178    2     3  1,0      1,0,0
%e A382178    3     3  1,1      1,1,1
%e A382178    4     3  2,0      2,0,0
%e A382178    5     3  2,1      2,1,1
%e A382178    6     4  1,0,0    1,0,0,0
%e A382178    7    18  1,0,1    1,0,1,0,0
%e A382178    8    18  1,1,0    1,1,0,0,0
%e A382178    9    17  1,1,1    1,1,1,1,1
%e A382178   10    17  1,2,0    1,2,0,1,0
%e A382178   11    16  1,2,1    1,2,1,1,0
%e A382178   12     4  2,0,0    2,0,0,0
%e A382178   13    19  2,0,1    2,0,1,0,1
%e A382178   14    19  2,1,0    2,1,0,1,0
%e A382178   15    18  2,1,1    2,1,1,0,0
%o A382178 (PARI) A153880(n) = { my (v = 0, f = 1); for (r = 2, oo, if (n==0, return (v);); v += (n%r) * f *= r; n \= r;); }
%o A382178 a(n) = { if (n==0, return (2)); my (m = n, f = 1, e); for (r = 2, oo, m = A153880(m); f *= r; e = (-m) % n; if (e < f, return ((m+e)/n););); }
%Y A382178 Cf. A153880, A382177.
%K A382178 nonn,base
%O A382178 0,1
%A A382178 _Rémy Sigrist_, Mar 17 2025