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These are the "first differences" between permutations of array A249821, in a sense that by composing the first k rows of this array [from left to right, as in a(n) = row_1(row_2(...(row_k(n))))], one obtains row k+1 of A249821.

On row n, the first A250473(n) terms are fixed, and the first non-fixed term comes at A250474(n).

These are the "first differences" between permutations of array A249822, in a sense that by composing the first k rows of this array [from right to left, as in a(n) = row_k(...(row_2(row_1(n))))], one obtains row k+1 of A249822.

On row n the first non-fixed term is A250474(n+1) at position A250474(n), i.e., on row 1 it is 5 at n=4, on row 2 it is 9 at n=5, on row 3 it is 14 at n=9, etc. All the previous A250473(n) terms are fixed.

a(n) = Length of the initial fixed portion of the permutation at row n of A251721 & A251722.

Also, for n > 1, the length of the maximal increasing prefix on row n of A249821.

Inverse of sequence A064216 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001

From Antti Karttunen, Dec 20 2014: (Start)

Permutation of natural numbers obtained by replacing each prime divisor of n with the next prime and mapping the generated odd numbers back to all natural numbers by adding one and then halving.

Note: there is a 7-cycle almost right in the beginning: (6 8 14 17 10 11 7). (See also comments at A249821. This 7-cycle is endlessly copied in permutations like A250249/A250250.)

The only 3-cycle in range 1 .. 402653184 is (2821 3460 5639).

For 1- and 2-cycles, see A245449.

(End)

The first 5-cycle is (1410, 2783, 2451, 2703, 2803). - Robert Israel, Jan 15 2015

From Michel Marcus, Aug 09 2020: (Start)

(5194, 5356, 6149, 8186, 10709), (46048, 51339, 87915, 102673, 137205) and (175811, 200924, 226175, 246397, 267838) are other 5-cycles.

(10242, 20479, 21413, 29245, 30275, 40354, 48241) is another 7-cycle. (End)

From Antti Karttunen, Feb 10 2021: (Start)

Somewhat artificially, also this permutation can be represented as a binary tree. Each child to the left is obtained by multiplying the parent by 3 and subtracting one, while each child to the right is obtained by applying A253888 to the parent:

1

|

................../ \..................

2 3

5......../ \........4 8......../ \........6

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

14 13 11 7 23 9 17 18

41 10 38 12 32 28 20 15 68 25 26 63 50 16 53 19

etc.

Each node's (> 1) parent can be obtained with A253889. Sequences A292243, A292244, A292245 and A292246 are constructed from the residues (mod 3) of the vertices encountered on the path from n to the root (1).

(End)

a((A003961(n) + 1) / 2) = n and A003961(a(n)) = 2*n - 1 for all n. If the sequence is indexed by odd numbers only, it becomes multiplicative. In this variant sequence, denoted b, even indices don't exist, and we get b(1) = a(1) = 1, b(3) = a(2) = 2, b(5) = 3, b(7) = 5, b(9) = 4 = b(3) * b(3), ... , b(15) = 6 = b(3) * b(5), and so on. This property can also be stated as: a(x) * a(y) = a(((2x - 1) * (2y - 1) + 1) / 2) for x, y > 0. - Reinhard Zumkeller [re-expressed by Peter Munn, May 23 2020]

Not multiplicative in usual sense - but letting m=2n-1=product_j (p_j)^(e_j) then a(n)=a((m+1)/2)=product_j (p_(j-1))^(e_j). - Henry Bottomley, Apr 15 2005

From Antti Karttunen, Jul 25 2016: (Start)

Several permutations that use prime shift operation A064989 in their definition yield a permutation obtained from their odd bisection when composed with this permutation from the right. For example, we have:

A243505(n) = A122111(a(n)).

A243065(n) = A241909(a(n)).

A244153(n) = A156552(a(n)).

A245611(n) = A243071(a(n)).

(End)

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

This is permutation of natural numbers larger than 1.

From Antti Karttunen, Dec 19 2014: (Start)

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.

For navigating in this array:

A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.

A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.

First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).

(End)

The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

Position of the first composite number (which is always 4) on row n of A249821. The fourth column of A249822.

Position of the first nonfixed term on row n of arrays of permutations A251721 and A251722.

According to the definition, this is the number of multiples of prime(n) below prime(n)^3 (and thus, the number of numbers below prime(n)^2) which do not have a smaller factor than prime(n). That is, the numbers remaining below prime(n)^2 after deleting all multiples of primes less than prime(n), as is done by applying the first n-1 steps of the sieve of Eratosthenes (when the first step is elimination of multiples of 2). This explains that the first differences are a(n+1)-a(n) = A050216(n)-1 for n>1, and a(n) = A054272(n)+2. - M. F. Hasler, Dec 31 2014