cp's OEIS Frontend

A384928 Number of 2-dense sublists of divisors of the n-th triangular number.

%I A384928 #35 Aug 14 2025 03:55:47
%S A384928 1,1,2,1,2,3,4,1,1,3,4,1,2,3,4,1,2,5,6,3,1,5,4,1,1,5,6,1,4,5,6,1,1,5,
%T A384928 6,1,2,3,6,1,2,7,8,3,1,3,4,1,1,5,6,3,4,7,3,1,1,5,4,1,2,3,8,1,1,7,8,3,
%U A384928 3,5,6,1,2,3,6,1,4,5,8,1,1,7,4,1,1,7,6,1,4,5,3,3,3,5,8,1,2,5,5,1,6
%N A384928 Number of 2-dense sublists of divisors of the n-th triangular number.
%C A384928 By definition a(n) is also the number of 2-dense sublists of divisors of the n-th generalized hexagonal number.
%C A384928 In a 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 A384928 The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
%C A384928 Conjecture: all odd indexed terms are odd.
%H A384928 Alois P. Heinz, <a href="/A384928/b384928.txt">Table of n, a(n) for n = 0..10000</a>
%F A384928 a(n) = A237271(A000217(n)) for n >= 1 (conjectured).
%e A384928 For n = 5 the 5th triangular number is 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(5) = 3.
%e A384928 For n = 12 the 12th triangular number is 78. The list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78], so a(12) = 2. Note that 78 is also the first practical number A005153 not in the sequence of the 2-dense numbers A174973.
%t A384928 A384928[n_] := Length[Split[Divisors[PolygonalNumber[n]], #2 <= 2*# &]];
%t A384928 Array[A384928, 100, 0] (* _Paolo Xausa_, Aug 14 2025 *)
%Y A384928 Cf. A000217, A005153, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386984 (a bisection), A386989.
%K A384928 nonn
%O A384928 0,3
%A A384928 _Omar E. Pol_, Aug 08 2025