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.

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

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]]*(k-1)!)))) }

Crossrefs

Cf. A007623, A306605.

Sequence in context: A215747 A246667 A199066 * A176948 A368624 A113138

Adjacent sequences: A306581 A306582 A306583 * A306585 A306586 A306587

Keyword

nonn,base

Author

Rémy Sigrist, Feb 25 2019

Status

approved