cp's OEIS Frontend

A pseudo-logarithmic function in the sense that a(b*c) = a(b)+a(c) and so a(b^c) = c*a(b) and f(n) = k^a(n) is a multiplicative function. [Cf. A248692 for example.] Essentially a function from the positive integers onto the partitions of the nonnegative integers (1->0, 2->1, 3->2, 4->1+1, 5->3, 6->1+2, etc.) so each value a(n) appears A000041(a(n)) times, first with the a(n)-th prime and last with the a(n)-th power of 2. Produces triangular numbers from primorials. - Henry Bottomley, Nov 22 2001

Michael Nyvang writes (May 08 2006) that the Danish composer Karl Aage Rasmussen discovered this sequence in the 1990's: it has excellent musical properties.

All A000041(a(n)) different n's with the same value a(n) are listed in row a(n) of triangle A215366. - Alois P. Heinz, Aug 09 2012

a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015

From Antti Karttunen, May 12 2014: (Start)

a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]

Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.

(End)

From Antti Karttunen, Dec 18-21 2014: (Start)

When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).

For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.

(End)

Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

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.)

(End)

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.

(End)

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

This permutation maps between the partitions as ordered in A112798 and A241918 (the original motivation for this sequence).

For all n > 2, A007814(a(n)) = A055396(n)-1, which implies that this self-inverse permutation maps between primes (A000040) and the powers of two larger than one (A000079(n>=1)), and apart from a(1) & a(2), this also maps each even number to some odd number, and vice versa, which means there are no fixed points after 2.

A122111 commutes with this one, that is, a(n) = A122111(a(A122111(n))).

Conjugates between A243051 and A242424 and other rows of A243060 and A243070.

In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them, which is added to the remaining set of piles. Essentially, this operation is a function whose domain and range are unordered integer partitions (cf. A000041) and which preserves the total size of a partition (the sum of its parts). This sequence is induced when the operation is implemented on the partitions as ordered by the list A112798.

Please compare to the definition of A122111, which conjugates the partitions encoded with the same system.

a(n) is even if and only if n is either a prime or a multiple of three.

Conversely, a(n) is odd if and only if n is a nonprime not divisible by three.

From Antti Karttunen, Jul 21 2014: (Start)

a(n) divides A122111(n), A242424(n), A243072(n), A243073(n) because a(n) divides all the terms in column n of A243070.

a(2n-1) divides A243505(n) and a(2n-1)^2 divides A122111(2n-1).

(End)

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

Each row is a A241909-conjugate of the corresponding row in A243070.

Rows in both arrays converge towards A122111.

First point where row k differs from row k of A243070 seems to be A000040(k+2): primes from five onward: 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

While the first point where row k differs from A122111 seems to begin as 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, ... A007283 (3*2^n) from the term n=1 onward.

The rows in this table preserve A243503.

The usual Bulgarian Solitaire operation (the "first order" version, cf. A242424) applied to an unordered integer partition means: subtract one from each part, and add a new part as large as there were parts in the old partition.

The "Second Order Bulgarian Solitaire" operation means that after subtracting one from each part of the old partition (and discarding the parts that diminished to zero), we apply the (first order) Bulgarian operation to the remaining partition before adding a new part as large as there were parts in the original partition.

In this context, where the parts of partitions are encoded with the indices of primes in the prime factorization of n (as in A112798), A064989(n) gives the remaining partition after one has been subtracted from each part; A242424 applies the first order Bulgarian operation to it; and multiplying with A000040(A001222(n)) adds a part as large as there originally were parts.

The usual (first-order) Bulgarian Solitaire operation (cf. A242424) applied to an unordered integer partition means: subtract one from each part, and add a new part as large as there were parts in the old partition.

The "Second-Order Bulgarian Operation" means that after subtracting one from each part of the old partition (and discarding the parts that diminished to zero), we apply the (first order) Bulgarian operation to the remaining partition before adding a new part as large as there were parts in the original partition.

Similarly, in "Third-Order Bulgarian Solitaire Operation", we apply the Second-Order Bulgarian operation to the remaining partition (after we have subtracted one from each part) before adding a new part as large as there were parts in the original partition.

In this context, where the parts of partitions are encoded with the indices of primes in the prime factorization of n (as in A112798), A064989(n) gives the remaining partition after one has been subtracted from each part; A243072 applies the second-order Bulgarian operation to it; and multiplying with A000040(A001222(n)) adds a part as large as there originally were parts.