

A306584


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.


2



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

0,4


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..5040
Wikipedia, Factorial number system
Index entries for sequences related to factorial base representation


FORMULA

a(n) <= n.
a(n) = A306605(n, n).


EXAMPLE

For n = 42:
 42 = 4! + 3*3!,
 f_42(0) = 0, f_42(f_42(0)) = 0,
 f_42(1) = 0, f_42(f_42(0)) = 0,
 f_42(2) = 0, f_42(f_42(2)) = 0,
 f_42(3) = 3, f_42(f_42(3)) = 3,
 f_42(4) = 1, f_42(f_42(4)) = 0,
 f_42(k) = 0, f_42(f_42(k)) = 0 for any k > 4,
 hence a(42) = 3*3! = 18.


PROG

(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]]*(k1)!)))) }


CROSSREFS

Cf. A007623, A306605.
Sequence in context: A215747 A246667 A199066 * A176948 A113138 A010263
Adjacent sequences: A306581 A306582 A306583 * A306585 A306586 A306587


KEYWORD

nonn,base


AUTHOR

Rémy Sigrist, Feb 25 2019


STATUS

approved



