cp's OEIS Frontend

Numbers k for which minimal number of factorials needed to add to get k is odd.

This sequence offers one possible analog to A000069 (odious numbers) in factorial base system. A227132 gives another kind of analog.

In each range [0,n!-1] exactly half of the integers are found in this sequence, and the other half of them are found in the complement, A227148.

The sequence gives the positions of odd permutations in the tables A055089 and A195663; and equivalently, the positions of odd numbers in A055091.

Equally: main diagonal of A261097.

For permutation p, which has rank n in permutation list A055089 (A195663), a(n) gives the rank of the "square" of that permutation (obtained by composing it with itself as: q(i) = p(p(i))) in the same list. Thus zeros (which mark the identity permutation, with rank 0) occur at positions where the permutations of A055089/A195663 are involutions, listed by A014489.

There are 5! = 120 terms in this finite subsequence of A030299.

It would be interesting to conceive simple and/or efficient functions which yield (a) the n-th term of this sequence: f(n) = a(n), (b) for a given term, the subsequent one: f(a(n)) = a(1 + (n mod 5!)).

From Nathaniel Johnston, May 19 2011: (Start)

Individual terms a(n) can be computed efficiently via the following procedure: Define b(n,k) = 1 + floor(((n-1) mod (k+1)!)/k!) for k = 1, 2, 3, 4. The first digit of a(n) is b(n,4). The second digit of a(n) is the b(n,3)-th number not already used. The third digit of a(n) is the b(n,2)-th number not already used. The fourth digit of a(n) is the b(n,1)-th number not already used, and the final digit of a(n) is the only digit remaining. This procedure generalizes in the obvious way for related sequences such as A178476.

For example, if n = 38 then we compute b(38,1) = 2, b(38,2) = 1, b(38,3) = 3, b(38,4) = 2. Thus a(38) = 24153 (2, followed by the 3rd digit not yet used, followed by the 1st digit not yet used, followed by the 2nd digit not yet used, followed by the last remaining digit).

(End)

Each row and column is a permutation of A001477. See the comments at A261096.

If the 120 permutations of S_5 are connected by adjacent transpositions, the graph produced is isomorphic to the prismatodecachoron (a 4-dimensional polytope) graph (see the Olshevsky link) and this sequence gives directions for a Hamiltonian circuit through its vertices. The first 24 terms give a Hamiltonian path through truncated octahedron's graph (the last path shown in the Karttunen link).

Comment from N. J. A. Sloane: This is the subject of "bell-ringing" or "change-ringing", which has been studied for hundreds of years. See for example Amer. Math. Monthly, Vol. 94, Number 8, 1987, pp. 721-.

This finite sequence contains 6!=720 terms.

This is a subsequence of A030299, consisting of elements A030299(154)..A030299(873).

If individual digits are be split up into separate terms, we get a subsequence of A030298.

It would be interesting to conceive simple and/or efficient functions which yield (a) the n-th term of this sequence: f(n)=a(n), (b) for a given term, the subsequent one: f(a(n)) = a(1 + (n mod 6!)).

The expression a(n+6) - a(n) takes only 18 different values for n = 1..6!-6.

An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011

From Hieronymus Fischer, Feb 13 2013: (Start)

The sum of all terms as decimal numbers is 279999720.

General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d different digits from the range 1..d < 10): sum = (d+1)!*(10^d-1)/18.

If the terms are interpreted as base-7 numbers the sum is 49412160.

General formula for the sum of all terms of the corresponding sequence of base-p permutational numbers (numbers with exactly p-1 different digits excluding the zero digit): sum = (p-2)!*(p^p-p)/2. (End)

There are 5!=120 terms in this finite sequence. Its origin is the fact that numbers whose decimal expansion is a permutation of 12345 are all of the form 9k+6.

The sequence is motivated by the fact that numbers whose decimal expansion is a permutation of 123456, are all of the form 9k+3.

There are 6!=720 terms in this finite sequence.

This sequence shows the cycle type of each finite permutation (A195663) as the index number of the corresponding partition. (When a permutation has a 3-cycle and a 2-cycle, this corresponds to the partition 3+2, etc.) Partitions can be ordered, so each partition can be denoted by its index in this order, e.g. 6 for the partition 3+2. Compare A194602.

From the properties of A194602 follows:

Entries 1,2,4,6,10,14,21... ( A000041(n)-1 from n=2 ) correspond to permutations with exactly one n-cycle (and no other cycles).

Entries 1,3,7,15,30,56,101... ( A000041(2n-1) from n=1 ) correspond to permutations with exactly n 2-cycles (and no other cycles), so these are the symmetric permutations.

Entries n = 1,3,4,7,9,10,12... ( A194602(n) has an even binary digit sum ) correspond to even permutations. This goes along with the fact, that a permutation is even when its partition contains an even number of even addends.

(Compare "Table for A194602" in section LINKS. Concerning the first two properties see especially the end of this file.)

Equally, for n>=1, each i in range [n!,(n+1)!-1] occurs n+1 times.

Used for computing A220659, A055089 and A060118: The n-th term a(n) tells which permutation (counted from the start, zero-based) of A055089 or A060117/A060118 the n-th term in those sequence belongs to.