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

%I A242424 #22 Mar 07 2017 13:16:25
%S A242424 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,
%T A242424 40,25,46,30,58,11,42,39,45,28,62,51,66,21,74,50,82,35,60,57,86,22,75,
%U A242424 45,78,55,94,56,63,35,102,69,106,42,118,87,100,13,99,70,122,65
%N A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).
%C A242424 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.
%C A242424 Please compare to the definition of A122111, which conjugates the partitions encoded with the same system.
%C A242424 a(n) is even if and only if n is either a prime or a multiple of three.
%C A242424 Conversely, a(n) is odd if and only if n is a nonprime not divisible by three.
%D A242424 Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.
%H A242424 Antti Karttunen, <a href="/A242424/b242424.txt">Table of n, a(n) for n = 1..8192</a>
%H A242424 Ethan Akin and Morton Davis, <a href="http://www.jstor.org/stable/2323643">"Bulgarian solitaire"</a>, American Mathematical Monthly 92 (4): 237-250. (1985).
%H A242424 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bulgarian_solitaire">Bulgarian solitaire</a>
%H A242424 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A242424 a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n) = A105560(n) * A064989(n).
%F A242424 a(n) = A241909(A243051(A241909(n))).
%F A242424 a(n) = A243353(A226062(A243354(n))).
%F A242424 a(A000079(n)) = A000040(n) for all n.
%F A242424 A056239(a(n)) = A056239(n) for all n.
%o A242424 (Scheme) (define (A242424 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A064989 n))))
%Y A242424 Row 1 of A243070 (table which gives successive "recursive iterates" of this sequence and converges towards A122111).
%Y A242424 Fixed points: A002110 (primorial numbers).
%Y A242424 Cf. A000041, A243051, A243072, A243073, A226062, A242422, A242423, A122111, A105560, A000040, A001222, A064989, A056239.
%K A242424 nonn
%O A242424 1,2
%A A242424 _Antti Karttunen_, May 13 2014