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a(n) = smallest positive multiple of n whose factorial base representation contains only 0's and 1's.
2

%I #12 Jun 26 2017 07:58:35

%S 1,2,3,8,25,6,7,8,9,30,33,24,26,126,30,32,153,126,152,120,126,726,

%T 5888,24,25,26,27,728,145,30,31,32,33,5066,840,144,5883,152,5070,120,

%U 123,126,129,5192,720,5888,752,144,147,150,153,728,848,864,46200,728

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

%C All terms belong to A059590.

%C a(n) = n iff n belongs to A059590.

%C 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 This sequence is to factorial base what A004290 is to decimal base.

%H Rémy Sigrist, <a href="/A286820/b286820.txt">Table of n, a(n) for n = 1..2000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system">Factorial number system</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%e The first terms are:

%e n a(n) a(n) in factorial base

%e -- ---- ----------------------

%e 1 1 1

%e 2 2 1,0

%e 3 3 1,1

%e 4 8 1,1,0

%e 5 25 1,0,0,1

%e 6 6 1,0,0

%e 7 7 1,0,1

%e 8 8 1,1,0

%e 9 9 1,1,1

%e 10 30 1,1,0,0

%e 11 33 1,1,1,1

%e 12 24 1,0,0,0

%e 13 26 1,0,1,0

%e 14 126 1,0,1,0,0

%e 15 30 1,1,0,0

%e 16 32 1,1,1,0

%e 17 153 1,1,1,1,1

%e 18 126 1,0,1,0,0

%e 19 152 1,1,1,1,0

%e 20 120 1,0,0,0,0

%o (PARI) isA059590(n) = my (r=2); while (n, if (n%r > 1, return (0), n\=r; r++)); return (1)

%o a(n) = forstep (m=n, oo, n, if (isA059590(m), return (m)))

%Y Cf. A004290, A007489, A059590, A284750.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Jun 24 2017