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A382177 a(n) is the least k > 1 such that the factorial base expansion of k*n starts with that of n while the remaining digits are zeros.

%I A382177 #8 Mar 20 2025 09:29:36
%S A382177 2,2,3,10,3,312,4,18,18,96,96,600,4,6168960,6120,18,18,11017036800,4,
%T A382177 56229997824000,114,760,68947200,18,5,14544,141120,192,13320,
%U A382177 9092075324665919034015350784000000,28,520412336961032355840000,27,1400,199584000,116496,180
%N A382177 a(n) is the least k > 1 such that the factorial base expansion of k*n starts with that of n while the remaining digits are zeros.
%C A382177 This sequence is well defined: for any n > 0 and m >= 0, A153880^m(n) (where A153880^m denotes the m-th iterate of A153880) is a multiple of (m+1)! whose factorial base expansion starts with that of n while the remaining digits are zeros, so for m sufficiently large, n will divide (m+1)! and hence this value.
%H A382177 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A382177 a(k!) = k+1 for any k > 0.
%e A382177 The first terms, in decimal and in factorial base, are:
%e A382177   n   a(n)     fact(n)  fact(a(n)*n)
%e A382177   --  -------  -------  ---------------------
%e A382177    0        2  0        0
%e A382177    1        2  1        1,0
%e A382177    2        3  1,0      1,0,0
%e A382177    3       10  1,1      1,1,0,0
%e A382177    4        3  2,0      2,0,0
%e A382177    5      312  2,1      2,1,0,0,0,0
%e A382177    6        4  1,0,0    1,0,0,0
%e A382177    7       18  1,0,1    1,0,1,0,0
%e A382177    8       18  1,1,0    1,1,0,0,0
%e A382177    9       96  1,1,1    1,1,1,0,0,0
%e A382177   10       96  1,2,0    1,2,0,0,0,0
%e A382177   11      600  1,2,1    1,2,1,0,0,0,0
%e A382177   12        4  2,0,0    2,0,0,0
%e A382177   13  6168960  2,0,1    2,0,1,0,0,0,0,0,0,0,0
%e A382177   14     6120  2,1,0    2,1,0,0,0,0,0,0
%e A382177   15       18  2,1,1    2,1,1,0,0
%o A382177 (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 A382177 a(n) = { my (m = n); while (1, m = A153880(m); if (m==0, return (2), m%n==0, return (m/n));); }
%Y A382177 Cf. A153880, A382178.
%K A382177 nonn,base
%O A382177 0,1
%A A382177 _Rémy Sigrist_, Mar 17 2025