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A381771 For any n > 0, a(n) is the least positive multiple of n whose factorial base expansion has digits in nonincreasing order; a(0) = 0.

%I A381771 #10 Mar 10 2025 11:11:27
%S A381771 0,1,2,3,4,5,6,14,8,9,20,22,12,65,14,15,16,17,18,57,20,21,22,23,24,
%T A381771 150,78,54,56,87,30,62,32,33,102,105,72,111,114,78,80,574,84,86,88,90,
%U A381771 92,94,48,294,150,102,104,424,54,110,56,57,116,118,60,305,62,63
%N A381771 For any n > 0, a(n) is the least positive multiple of n whose factorial base expansion has digits in nonincreasing order; a(0) = 0.
%C A381771 The sequence is well defined as for any n > 0, the factorial base expansion of n! has digits in nonincreasing order.
%H A381771 Rémy Sigrist, <a href="/A381771/b381771.txt">Table of n, a(n) for n = 0..10000</a>
%H A381771 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A381771 a(n) = A381770(n) * n.
%F A381771 a(n) <= n!.
%F A381771 a(n) = n iff n belongs to A351987.
%e A381771 The first terms, alongside their factorial base expansion, are:
%e A381771   n   a(n)  fact(a(n))
%e A381771   --  ----  ----------
%e A381771    0     0  0
%e A381771    1     1  1
%e A381771    2     2  1,0
%e A381771    3     3  1,1
%e A381771    4     4  2,0
%e A381771    5     5  2,1
%e A381771    6     6  1,0,0
%e A381771    7    14  2,1,0
%e A381771    8     8  1,1,0
%e A381771    9     9  1,1,1
%e A381771   10    20  3,1,0
%e A381771   11    22  3,2,0
%e A381771   12    12  2,0,0
%e A381771   13    65  2,2,2,1
%e A381771   14    14  2,1,0
%e A381771   15    15  2,1,1
%o A381771 (PARI) is(n) = { my (p = -1); for (r = 2, oo, if (n==0, return (1), p > p = n%r, return (0)); n \= r;); }
%o A381771 a(n) = { for (k = 1, oo, if (is(k*n), return (k*n););); }
%o A381771 (Python)
%o A381771 from itertools import count
%o A381771 def facbase(n, i=2): return [n] if n < i else [*facbase(n//i, i=i+1), n%i]
%o A381771 def is_non_inc(n): return (fb:=facbase(n)) == sorted(fb, reverse=True)
%o A381771 def a(n): return next(k*n for k in count(1) if is_non_inc(k*n))
%o A381771 print([a(n) for n in range(64)]) # _Michael S. Branicky_, Mar 09 2025
%Y A381771 Cf. A223474, A351987, A381770.
%K A381771 nonn,base
%O A381771 0,3
%A A381771 _Rémy Sigrist_, Mar 07 2025