A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12, 13, 35, 10, 16, 17, 18, 19, 45, 21, 77, 23, 24, 25, 143, 27, 175, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 49, 75, 187, 1573, 53, 54, 33, 875, 247, 667, 59, 90, 61, 899, 63, 64, 65

Offset: 1

Author

Antti Karttunen, May 31 2014

Keywords

• nonn

Comments

This self-inverse permutation (involution) of positive integers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].

It also preserves the exponent of the largest prime factor (A071178), from which follows that the sequence A102750 is closed with respect to this permutation, i.e., for all n in A102750, a(n) is either same n or some other term of A102750.

Considered as an operation on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements a self-inverse bijection which is a composition of the effects of A242419 and A225891. (Or equally: A105119 and A242419). For details, please see the respective Comments sections and/or Example section of this entry.

Examples

			For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
First we apply A242419, which reverses the order of "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, so that the first stair from the top (one unit wide and three units high) is moved to the last position, the second one (two units wide and two units high) stays in the middle, and the original bottom step (two units wide and one unit high) will be the new topmost step, thus we get the following Young diagram:
   _ _
  |   |_ _
  |       |
  |       |_
  |         |
  |         |
  |_ _ _ _ _|
which represents the partition (2,4,4,5,5,5), encoded in A112798 by p_2 * p_4^2 * p_5^3 = 3 * 7^2 * 11^3 = 195657.
Then we apply A225891, which rotates the exponents of distinct primes in the factorization of n one left, in this context the vertical line segments one step up, with the top-one going to the bottomost, and so we get:
   _ _
  |   |
  |   |_ _
  |       |
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,4,4,4,5), encoded in A112798 by p_2^2 * p_4^3 * p_5 = 3^2 * 7^3 * 11 = 33957, thus a(2200) = 33957.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• Wikipedia, Young diagram
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Fixed points: A243068.

Cf. A112798, A242419, A105119, A225891, A006530, A071178, A102750, A243057, A243059.

Programs

Scheme

(define (A242420 n) (* (expt (A006530 n) (- (A071178 n) 1)) (A243057 n)))

Formula

a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

For all k in A102750, a(k) = A243057(k) = A243059(k).

By composing related permutations:

a(n) = A225891(A242419(n)) = A242419(A105119(n)).