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a(n) = least k > 0 such that k * n in factorial base representation contains only 0's and 1's.
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%I #43 Jun 26 2017 07:58:22

%S 1,1,1,2,5,1,1,1,1,3,3,2,2,9,2,2,9,7,8,6,6,33,256,1,1,1,1,26,5,1,1,1,

%T 1,149,24,4,159,4,130,3,3,3,3,118,16,128,16,3,3,3,3,14,16,16,840,13,

%U 89,15,88,2,2,12,2,2,78,11,13,76,597,12,71,2,2,555,2

%N a(n) = least k > 0 such that k * n in factorial base representation contains only 0's and 1's.

%C a(n) = 1 iff n belongs to A059590.

%C a(a(n)) <= n for any n > 0.

%H Rémy Sigrist, <a href="/A284750/b284750.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>

%F a(n) = A286820(n) / n.

%e The first terms, alongside n*a(n) in decimal and factorial base, are:

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

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

%e 1 1 1 1

%e 2 1 2 1,0

%e 3 1 3 1,1

%e 4 2 8 1,1,0

%e 5 5 25 1,0,0,1

%e 6 1 6 1,0,0

%e 7 1 7 1,0,1

%e 8 1 8 1,1,0

%e 9 1 9 1,1,1

%e 10 3 30 1,1,0,0

%e 11 3 33 1,1,1,1

%e 12 2 24 1,0,0,0

%e 13 2 26 1,0,1,0

%e 14 9 126 1,0,1,0,0

%e 15 2 30 1,1,0,0

%e 16 2 32 1,1,1,0

%e 17 9 153 1,1,1,1,1

%e 18 7 126 1,0,1,0,0

%e 19 8 152 1,1,1,1,0

%e 20 6 120 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) = for (k=1, oo, if (isA059590(k*n), return (k)))

%Y Cf. A059590, A286820.

%K nonn,base

%O 1,4

%A _Rémy Sigrist_, Jun 25 2017