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A384931 Number of 2-dense sublists of divisors of the number of partitions of n.

%I A384931 #24 Aug 28 2025 10:25:07
%S A384931 1,1,1,2,2,2,2,3,2,1,1,1,3,2,3,1,5,6,4,4,5,1,2,4,3,4,1,5,4,7,2,4,9,10,
%T A384931 4,9,2,6,9,3,1,9,4,11,8,4,3,3,8,12,4,11,7,10,5,3,7,2,2,1,8,5,6,8,5,2,
%U A384931 1,3,10,6,1,6,8,7,1,1,4,2,7,9,3,4,9,6,2
%N A384931 Number of 2-dense sublists of divisors of the number of partitions of n.
%C A384931 In a 2-dense sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
%C A384931 The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
%H A384931 Alois P. Heinz, <a href="/A384931/b384931.txt">Table of n, a(n) for n = 0..4000</a>
%F A384931 a(n) = A237271(A000041(n)). Conjectured.
%e A384931 For n = 7 the number of partitions of 7 is A000041(7) = 15. The list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15], so a(7) = 3.
%e A384931 For n = 19 the number of partitions of 19 is A000041(19) = 490. The list of divisors of 490 is [1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]. There are four 2-dense sublists of divisors of 490, they are [1, 2], [5, 7, 10, 14], [35, 49, 70, 98], [245, 490], so a(19) = 4.
%t A384931 A384931[n_] := Length[Split[Divisors[PartitionsP[n]], #2 <= 2*# &]];
%t A384931 Array[A384931, 100, 0] (* _Paolo Xausa_, Aug 28 2025 *)
%Y A384931 Cf. A000041, A174973 (2-dense numbers), A237271, A384149, A384222, A384225, A384226, A384930.
%K A384931 nonn,changed
%O A384931 0,4
%A A384931 _Omar E. Pol_, Jul 30 2025
%E A384931 More terms from _Alois P. Heinz_, Jul 30 2025