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. 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
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..2455
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Wikipedia, Bulgarian solitaire
EXAMPLE
1 is present as it has an empty factorization, for which the sum of prime indices is zero, and zero is also a triangular number.
2 = p_1 is present as 1 is a triangular number.
6 = p_1 * p_2 is present, as 1+2 = 3 is a triangular number.
300 = 2*2*3*5*5 = p_1 * p_1 * p_2 * p_3 * p_3 is present, as 1+1+2+3+3 = 10 is a triangular number.
Any primorial number p_1 * p_2 * p_3 * ... * p_n is present, as 1+2+3+...+n is by definition a triangular number.
The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (), (1), (3), (2,1), (1,1,1), (6), (4,2), (5,1), (3,3), (2,2,2), (4,1,1), (10), (3,2,1), (2,2,1,1), (3,1,1,1), (9,1), (15), (2,1,1,1,1), (8,2), (1,1,1,1,1,1), (21), (8,1,1), (7,3), (14,1), (6,4). - Gus Wiseman, Nov 13 2018
MATHEMATICA
triQ[n_]:=Module[{k, i}, For[k=n; i=1, k>0, i++, k-=i]; k==0];
Select[Range[100], triQ[Total[Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]*k]]]&] (* Gus Wiseman, Nov 13 2018 *)
PROG
(Scheme, with Antti Karttunen's Intseq-library)
KEYWORD
nonn
AUTHOR
Antti Karttunen, May 16 2014
STATUS
approved