A387909 Consider the factorial base expansion of n, without leading zeros, say (d(1), ..., d(w)); for k = 2..floor(w/2), if d(k) <= k then exchange d(k) and d(w+1-k); the resulting digits correspond to the factorial base expansion of a(n).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 31, 36, 37, 26, 27, 32, 33, 38, 39, 28, 29, 34, 35, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54, 55, 60, 61, 50, 51, 56, 57, 62, 63, 52, 53, 58, 59, 64, 65, 66, 67

Offset

0,3

Comments

A self-inverse permutation of the nonnegative integers that preserves the number of digits (A084558) and the sum of digits (A034968) in factorial base.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..5040

Index entries for sequences related to factorial base representation

Index entries for sequences that are permutations of the natural numbers

Formula

a(a(n)) = n.

Example

For n = 1361: the factorial base expansion of 1361 is (d(1), ..., d(6)) = (1,5,1,2,2,1); d(2) = 5 > 2 so we don't alter d(2) and d(5); d(3) = 1 <= 3 so we exchange d(3) and d(4); the resulting digits, (1,5,2,1,2,1), correspond to 1379, so a(1361) = 1379.

Prog

(PARI) factdigits(n) = { my (dd=[]); for (r=2, oo, if (n==0, return (dd), dd=concat(n%r, dd); n\=r; ); ); }
fromfactdigits(d) = { sum (k=1, #d, d[#d+1-k]*k!); }
a(n) = {
	my (d = factdigits(n));
	for (k = 2, #d\2, if (d[k] <= k, [d[k], d[#d+1-k]] = [d[#d+1-k], d[k]]; ); );
	fromfactdigits(d);
}

Crossrefs

Cf. A034968, A084558, A333658, A389644.

Sequence in context: A243903 A391014 A296867 * A228144 A236841 A236850

Adjacent sequences: A387906 A387907 A387908 * A387910 A387911 A387912

Keyword

nonn,base

Author

Rémy Sigrist, Oct 12 2025

Status

approved