cp's OEIS Frontend

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)].

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.

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:

A001222: gives the height of whole diagram,

A051119: removes the bottommost step from the diagram,

A241919: gives the length of the horizontal line segment of the bottom step, i.e. its width,

A071178: gives the length of the vertical line segment of the bottom step, i.e. its height,

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,

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.

Number n is present if its prime factorization n = p_a * p_b * p_c * ... * p_i * p_j * p_k (where a <= b <= c <= ... <= i <= j <= k are the indices of prime factors, not necessarily all distinct; sorted into nondescending order) satisfies the condition that the first differences of those prime indices (a-0, b-a, c-b, ..., j-i, k-j) form a palindrome.

The above condition implies that none of the terms of A070003 are present, as then at least the difference k-j would be zero, but on the other hand, a-0 is at least 1. Cf. also A243068.

Number n is present, if its prime factorization n = p_a^e_a * p_b^e_b * p_c^e_c * ... * p_i^e_i * p_j^e_j * p_k^e_k where a < b < c < ... < i < j < k, satisfies the condition that the first differences of prime indices (a-0, b-a, c-b, ..., j-i, k-j) form a palindrome.

More formally, numbers n whose prime factorization is either of the form p^e (p prime, e >= 0), i.e., one of the terms of A000961, or of the form p_i1^e_i1 * p_i2^e_i2 * p_i3^e_i3 * ... * p_i_{k-1}^e_{i_{k-1}} * p_{i_k}^e_{i_k}, where p_i1 < p_i2 < ... < p_i_{k-1} < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_i1 .. e_{i_k} are their nonzero exponents (here k = A001221(n) and i_k = A061395(n), the index of the largest prime present), and the indices of primes satisfy the relation that for each index i_j < i_k present, the index i_{k-j} is also present.

A number n is present if its prime factorization n = p_a * p_b * p_c * ... * p_i * p_j * p_k^e_k, where a <= b <= c <= ... <= i <= j < k are the indices of prime factors, not necessarily all distinct, except that j < k, and the greatest prime divisor p_k [with k = A061395(n)] may occur multiple times, satisfies the condition that the first differences of those prime indices (a-0, b-a, c-b, ..., j-i, k-j) form a palindrome.