%I #11 Jul 12 2019 17:07:29

%S 0,1,0,3,4,5,0,7,0,9,4,11,0,1,6,9,16,17,18,19,18,21,22,23,0,25,0,27,4,

%T 29,0,31,0,33,4,35,0,25,6,33,16,41,18,43,18,45,22,47,0,1,24,27,52,53,

%U 0,7,24,33,52,59,0,1,30,33,64,65,18,19,42,45,70,71,0

%N For any n >= 0, let f_n be the representation of n in the factorial number system: for any i >= 0, 0 <= f_n(i) <= i and n = Sum_{i >= 0} f_n(i) * i!. The representation of a(n) in the factorial number system, say g, satisfies g(i) = f_n(f_n(i)) for any i >= 0.

%H Rémy Sigrist, <a href="/A306584/b306584.txt">Table of n, a(n) for n = 0..5040</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) <= n.

%F a(n) = A306605(n, n).

%e For n = 42:

%e - 42 = 4! + 3*3!,

%e - f_42(0) = 0, f_42(f_42(0)) = 0,

%e - f_42(1) = 0, f_42(f_42(0)) = 0,

%e - f_42(2) = 0, f_42(f_42(2)) = 0,

%e - f_42(3) = 3, f_42(f_42(3)) = 3,

%e - f_42(4) = 1, f_42(f_42(4)) = 0,

%e - f_42(k) = 0, f_42(f_42(k)) = 0 for any k > 4,

%e - hence a(42) = 3*3! = 18.

%o (PARI) a(n) = { my (f=[]); for (r=1, oo, f = concat(f, n%r); n \= r; if (n==0, return (sum(k=1, #f, f[1+f[k]]*(k-1)!)))) }

%Y Cf. A007623, A306605.

%K nonn,base

%O 0,4

%A _Rémy Sigrist_, Feb 25 2019