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A286820 a(n) = smallest positive multiple of n whose factorial base representation contains only 0's and 1's.

%I A286820 #12 Jun 26 2017 07:58:35
%S A286820 1,2,3,8,25,6,7,8,9,30,33,24,26,126,30,32,153,126,152,120,126,726,
%T A286820 5888,24,25,26,27,728,145,30,31,32,33,5066,840,144,5883,152,5070,120,
%U A286820 123,126,129,5192,720,5888,752,144,147,150,153,728,848,864,46200,728
%N A286820 a(n) = smallest positive multiple of n whose factorial base representation contains only 0's and 1's.
%C A286820 All terms belong to A059590.
%C A286820 a(n) = n iff n belongs to A059590.
%C A286820 The sequence is well defined: for any n > 0: according to the pigeonhole principle, among the n+1 first repunits in factorial base (A007489), there must be two distinct terms equal modulo n; their absolute difference is a positive multiple of n, and contains only 0's and 1's in factorial base.
%C A286820 This sequence is to factorial base what A004290 is to decimal base.
%H A286820 Rémy Sigrist, <a href="/A286820/b286820.txt">Table of n, a(n) for n = 1..2000</a>
%H A286820 Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system">Factorial number system</a>
%H A286820 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%e A286820 The first terms are:
%e A286820 n   a(n)     a(n) in factorial base
%e A286820 --  ----     ----------------------
%e A286820 1   1        1
%e A286820 2   2        1,0
%e A286820 3   3        1,1
%e A286820 4   8        1,1,0
%e A286820 5   25       1,0,0,1
%e A286820 6   6        1,0,0
%e A286820 7   7        1,0,1
%e A286820 8   8        1,1,0
%e A286820 9   9        1,1,1
%e A286820 10  30       1,1,0,0
%e A286820 11  33       1,1,1,1
%e A286820 12  24       1,0,0,0
%e A286820 13  26       1,0,1,0
%e A286820 14  126      1,0,1,0,0
%e A286820 15  30       1,1,0,0
%e A286820 16  32       1,1,1,0
%e A286820 17  153      1,1,1,1,1
%e A286820 18  126      1,0,1,0,0
%e A286820 19  152      1,1,1,1,0
%e A286820 20  120      1,0,0,0,0
%o A286820 (PARI) isA059590(n) = my (r=2); while (n, if (n%r > 1, return (0), n\=r; r++)); return (1)
%o A286820 a(n) = forstep (m=n, oo, n, if (isA059590(m), return (m)))
%Y A286820 Cf. A004290, A007489, A059590, A284750.
%K A286820 nonn,base
%O A286820 1,2
%A A286820 _Rémy Sigrist_, Jun 24 2017