cp's OEIS Frontend

A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.

%I A242422 #18 Nov 14 2018 01:37:22
%S A242422 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 A242422 91,102,107,117,121,123,130,136,142,147,151,154,156,164,165,175,185,
%U A242422 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 A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.
%C A242422 Numbers k such that A010054(A056239(k)) is one, or equally, that A002262(A056239(k)) is zero.
%C A242422 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 A242422 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 A242422 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 A242422 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 A242422 Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.
%H A242422 Antti Karttunen, <a href="/A242422/b242422.txt">Table of n, a(n) for n = 1..2455</a>
%H A242422 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 A242422 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bulgarian_solitaire">Bulgarian solitaire</a>
%e A242422 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 A242422 2 = p_1 is present as 1 is a triangular number.
%e A242422 6 = p_1 * p_2 is present, as 1+2 = 3 is a triangular number.
%e A242422 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 A242422 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 A242422 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 A242422 triQ[n_]:=Module[{k,i},For[k=n;i=1,k>0,i++,k-=i];k==0];
%t A242422 Select[Range[100],triQ[Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]]&] (* _Gus Wiseman_, Nov 13 2018 *)
%o A242422 (Scheme, with _Antti Karttunen_'s Intseq-library)
%o A242422 (define A242422 (NONZERO-POS 1 1 (COMPOSE A010054 A056239)))
%Y A242422 Complement: A242423.
%Y A242422 A002110 (primorial numbers) is a subsequence.
%Y A242422 Cf. A112798, A215366, A242424, A000217, A002262, A010054, A056239, A226062, A037481.
%Y A242422 Cf. A066723, A321470, A321471, A321472.
%K A242422 nonn
%O A242422 1,2
%A A242422 _Antti Karttunen_, May 16 2014