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A242419 Reverse both the exponents and the deltas of the indices of distinct primes present in the prime factorization of n.

%I A242419 #37 Jun 02 2014 22:23:05
%S A242419 1,2,3,4,5,6,7,8,9,15,11,18,13,35,10,16,17,12,19,75,21,77,23,54,25,
%T A242419 143,27,245,29,30,31,32,55,221,14,36,37,323,91,375,41,105,43,847,50,
%U A242419 437,47,162,49,45,187,1859,53,24,33,1715,247,667,59,150,61,899,147,64,65
%N A242419 Reverse both the exponents and the deltas of the indices of distinct primes present in the prime factorization of n.
%C A242419 This self-inverse permutation (involution) of natural numbers 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)].
%C A242419 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 bijection which reverses the order of "steps" in Young (or Ferrers) diagram of a partition (but keeps the horizontal line segment of each step horizontal and the vertical line segment vertical). Please see the last example in the example section.
%C A242419 To understand the given recursive formula, it helps to see that in the above context (Young diagrams drawn with French notation), the sequences employed effect the following operations:
%C A242419   A001222: gives the height of whole diagram,
%C A242419   A051119: removes the bottommost step from the diagram,
%C A242419   A241919: gives the length of the horizontal line segment of the bottom step, i.e. its width,
%C A242419   A071178: gives the length of the vertical line segment of the bottom step, i.e. its height,
%C A242419   A242378(k,n): increases the width of whole Young diagram encoded by n by adding a rectangular area A001222(n) squares high and k squares wide to its left,
%C A242419   and finally, multiplying by A000040(a)^b adds a new topmost step whose width is a and height is b. Particularly, multiplying by (A000040(A241919(n))^A071178(n)) transfers the bottommost step to the top.
%H A242419 Antti Karttunen, <a href="/A242419/b242419.txt">Table of n, a(n) for n = 1..8192</a>
%H A242419 Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_diagram">Young diagram</a>
%H A242419 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A242419 If n = p_a^e_a * p_b^e_b * ... * p_h^e_h * p_i^e_i * p_j^e_j * p_k^e_k, where p_a < ... < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_a .. e_k are their nonzero exponents, then a(n) = p_{k-j}^e_k * p_{k-i}^e_j  * p_{k-h}^e_i * ... * p_{k-a}^e_b * p_k^e_a.
%F A242419 As a recurrence:
%F A242419 a(1) = 1, and for n>1, a(n) = (A000040(A241919(n))^A071178(n)) * A242378(A241919(n), a(A051119(n))).
%F A242419 By composing related permutations:
%F A242419 a(n) = A122111(A153212(n)) = A153212(A122111(n)).
%F A242419 a(n) = A069799(A242415(n)) = A242415(A069799(n)).
%F A242419 a(n) = A105119(A242420(n)).
%e A242419 For n = 10 = 2*5 = p_1^1 * p_3^1, we get p_(3-1)^1 * p_3^1 = 3 * 5 = 15, thus a(10) = 15.
%e A242419 For n = 20 = 2*2*5 = p_1^2 * p_3^1, we get p_(3-1)^1 * p_3^2 = 3^1 * 5^2 = 3*25 = 75, thus a(20) = 75.
%e A242419 For n = 84 = 2*2*3*7 = p_1^2 * p_2 * p_4, when we reverse the deltas of indices, and reverse also the order of exponents, we get p_(4-2) * p_(4-1) * p_4^2 = 3 * 5 * 7^2 = 735, thus a(84) = 735.
%e A242419 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:
%e A242419    _
%e A242419   | |
%e A242419   | |
%e A242419   | |_ _
%e A242419   |     |
%e A242419   |     |_ _
%e A242419   |_ _ _ _ _|
%e A242419 Reversing the order of "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, means 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:
%e A242419    _ _
%e A242419   |   |_ _
%e A242419   |       |
%e A242419   |       |_
%e A242419   |         |
%e A242419   |         |
%e A242419   |_ _ _ _ _|
%e A242419 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, thus a(2200) = 195657.
%o A242419 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A242419 (definec (A242419 n) (cond ((<= n 1) n) (else (* (expt (A000040 (A241919 n)) (A071178 n)) (A242378bi (A241919 n) (A242419 (A051119 n)))))))
%Y A242419 Fixed points: A242417.
%Y A242419 Cf. A112798, A122111, A153212, A000040, A241919, A071178, A242378, A051119, A243057, A105119, A242420.
%Y A242419 {A000027, A122111, A153212, A242419} form a 4-group.
%Y A242419 {A000027, A069799, A242415, A242419} form also a 4-group.
%K A242419 nonn
%O A242419 1,2
%A A242419 _Antti Karttunen_, May 17 2014