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Numbers in whose prime factorization the indices of primes sum to a triangular number.
11

%I #18 Nov 14 2018 01:37:22

%S 1,2,5,6,8,13,21,22,25,27,28,29,30,36,40,46,47,48,57,64,73,76,85,86,

%T 91,102,107,117,121,123,130,136,142,147,151,154,156,164,165,175,185,

%U 189,196,197,198,201,206,208,210,217,220,222,225,243,250,252,257,264,268,270,279,280,296,298,299,300

%N Numbers in whose prime factorization the indices of primes sum to a triangular number.

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

%C 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.

%C 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.

%C 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.

%C 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

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

%H Antti Karttunen, <a href="/A242422/b242422.txt">Table of n, a(n) for n = 1..2455</a>

%H 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bulgarian_solitaire">Bulgarian solitaire</a>

%e 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.

%e 2 = p_1 is present as 1 is a triangular number.

%e 6 = p_1 * p_2 is present, as 1+2 = 3 is a triangular number.

%e 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.

%e Any primorial number p_1 * p_2 * p_3 * ... * p_n is present, as 1+2+3+...+n is by definition a triangular number.

%e 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

%t triQ[n_]:=Module[{k,i},For[k=n;i=1,k>0,i++,k-=i];k==0];

%t Select[Range[100],triQ[Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]]&] (* _Gus Wiseman_, Nov 13 2018 *)

%o (Scheme, with _Antti Karttunen_'s Intseq-library)

%o (define A242422 (NONZERO-POS 1 1 (COMPOSE A010054 A056239)))

%Y Complement: A242423.

%Y A002110 (primorial numbers) is a subsequence.

%Y Cf. A112798, A215366, A242424, A000217, A002262, A010054, A056239, A226062, A037481.

%Y Cf. A066723, A321470, A321471, A321472.

%K nonn

%O 1,2

%A _Antti Karttunen_, May 16 2014