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

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.

Numbers k such that A010054(A056239(k)) is one, or equally, that A002262(A056239(k)) is zero.

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. The question originally posed was: on what condition the resulting partitions will eventually reach a fixed point, that is, a collection of piles that will be unchanged by the operation. See Martin Gardner reference and the Wikipedia-page.

This sequence answers the question when we implement the operation on the partition list A112798: These are all such numbers that starting iterating A242424 from them leads eventually to a fixed point, which will be one of the primorial numbers, A002110.

Contains the same terms as rows of A215366 indexed with triangular numbers (A000217: 0, 1, 3, 6, ...), although not in the same order. {1}, {2}, {5, 6, 8}, {13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64}, etc.

Heinz numbers of integer partitions of triangular numbers. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). - Gus Wiseman, Nov 13 2018