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

1,2

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

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.

Prog

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

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.

Sequence in context: A235451 A348915 A039819 * A232271 A194277 A226246

Adjacent sequences: A242421 A242422 A242423 * A242425 A242426 A242427

Keyword

nonn

Author

Antti Karttunen, May 13 2014

Status

approved