A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

1, 2, 4, 3, 6, 6, 10, 5, 12, 9, 14, 10, 22, 15, 18, 7, 26, 20, 34, 15, 30, 21, 38, 14, 27, 33, 40, 25, 46, 30, 58, 11, 42, 39, 45, 28, 62, 51, 66, 21, 74, 50, 82, 35, 60, 57, 86, 22, 75, 45, 78, 55, 94, 56, 63, 35, 102, 69, 106, 42, 118, 87, 100, 13, 99, 70, 122, 65

Offset: 1

Author

Antti Karttunen, May 13 2014

Keywords

• nonn

Comments

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.

References

• Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
• Wikipedia, Bulgarian solitaire
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Row 1 of A243070 (table which gives successive "recursive iterates" of this sequence and converges towards A122111).

Fixed points: A002110 (primorial numbers).

Cf. A000041, A243051, A243072, A243073, A226062, A242422, A242423, A122111, A105560, A000040, A001222, A064989, A056239.

Programs

Scheme

(define (A242424 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A064989 n))))

Formula

a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n) = A105560(n) * A064989(n).

a(n) = A241909(A243051(A241909(n))).

a(n) = A243353(A226062(A243354(n))).

a(A000079(n)) = A000040(n) for all n.

A056239(a(n)) = A056239(n) for all n.