cp's OEIS Frontend

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.

Factor n; replace each prime(i) with i, take the conjugate partition, replace parts i with prime(i) and multiply out.

From Antti Karttunen, May 12-19 2014: (Start)

For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).

Because the partition conjugation doesn't change the partition's total sum, this permutation preserves A056239, i.e., A056239(a(n)) = A056239(n) for all n.

(Similarly, for all n, A001221(a(n)) = A001221(n), because the number of steps in the Ferrers/Young-diagram stays invariant under the conjugation. - Note added Apr 29 2022).

Because this permutation commutes with A241909, in other words, as a(A241909(n)) = A241909(a(n)) for all n, from which follows, because both permutations are self-inverse, that a(n) = A241909(a(A241909(n))), it means that this is also induced when partitions are conjugated in the partition enumeration system A241918. (Not only in A112798.)

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From Antti Karttunen, Jul 31 2014: (Start)

Rows in arrays A243060 and A243070 converge towards this sequence, and also, assuming no surprises at the rate of that convergence, this sequence occurs also as the central diagonal of both.

Each even number is mapped to a unique term of A102750 and vice versa.

Conversely, each odd number (larger than 1) is mapped to a unique term of A070003, and vice versa. The permutation pair A243287-A243288 has the same property. This is also used to induce the permutations A244981-A244984.

Taking the odd bisection and dividing out the largest prime factor results in the permutation A243505.

Shares with A245613 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

Conversely, with A245614 (the inverse of above), shares the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.

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The Maple program follows the steps described in the first comment. The subprogram C yields the conjugate partition of a given partition. - Emeric Deutsch, May 09 2015

The Heinz number of the partition that is conjugate to the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r). Example: a(3) = 4. Indeed, the partition with Heinz number 3 is [2]; its conjugate is [1,1] having Heinz number 4. - Emeric Deutsch, May 19 2015

Equivalent description nearer to the old name: a(n) is a number obtained by reversing the indices of the primes present in the prime factorization of n, from the smallest to the largest, while keeping the nonzero exponents of those same primes at their old positions.

This self-inverse permutation of natural numbers fixes the numbers in whose prime factorization the sequence of nonzero exponents form a palindrome: A242414.

Integers which are changed are A242416.

Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which reverses the order of vertical line segments of the "steps" in Young (or Ferrers) diagram of a partition, but keeps the order of horizontal line segments intact. Please see the last example in the example section.

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.

The terms in the same column have the same prime signature.

The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

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.

In order for the "index difference" to make sense, we consider the factorization to be sorted with respect to the primes but not the powers to which they are raised; that is, first comes the smallest prime and each subsequent prime is larger than the previous disregarding their powers.

For every n it is true that a(a(n)) = n.

From Antti Karttunen, May 29 2014: (Start)

In other words, this is a self-inverse permutation (involution) of natural numbers.

This permutation maps primes (A000040) to the powers of two larger than one (A000079(n>=1)) and vice versa.

The term a(1) = 1 was added on the grounds that as 1 has an empty prime factorization, there is nothing to swap, thus it stays same. It is also needed as a base case for the given recurrence.

Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which exchanges the horizontal and vertical line segment of each "step" in Young (or Ferrers) diagram of a partition. Please see the last example in the example section.

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Numbers that are fixed by the permutation A242419, i.e., for which A242419(n) = n. Also, numbers that are fixed by both A069799 and A242415.

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 both the first differences of prime indices (a-0, b-a, c-b, ..., j-i, k-j) and the respective exponents (e_a, e_b, e_c, ... , e_i, e_j, e_k) 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_1 <= i_j < i_k present, the index i_{k-j} is also present, and the exponents e_{i_j} and e_{i_{(k-j)+1}} are equal.