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A386994 Number of 2-dense sublists of divisors of the n-th Fibonacci number.

%I A386994 #21 Sep 02 2025 05:58:47
%S A386994 1,1,1,1,2,2,1,2,4,2,4,2,1,2,4,4,8,2,3,4,8,4,4,2,1,6,4,4,12,2,1,4,16,
%T A386994 4,4,8,1,8,8,4,3,4,1,2,11,6,8,2,1,8,10,4,12,4,3,13,5,10,8,4,1,4,8,10,
%U A386994 17,8,7,8,20,9,15,4,1,4,16,18,24,15,7,4,3,5
%N A386994 Number of 2-dense sublists of divisors of the n-th Fibonacci number.
%C A386994 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 A386994 The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
%F A386994 a(n) = A237271(A000045(n)), n >= 1. (conjectured).
%e A386994 For n = 18 the 18th Fibonacci number is 2584. The list of divisors of 2584 is [1, 2, 4, 8, 17, 19, 34, 38, 68, 76, 136, 152, 323, 646, 1292, 2584]. There are three 2-dense sublists of divisors of 2584, they are [1, 2, 4, 8], [17, 19, 34, 38, 68, 76, 136, 152] and [323, 646, 1292, 2584], so a(18) = 3.
%t A386994 A386994[n_] := Length[Split[Divisors[Fibonacci[n]], #2 <= 2*# &]];
%t A386994 Array[A386994, 100, 0] (* _Paolo Xausa_, Sep 02 2025 *)
%Y A386994 Cf. A000045, A133021, A139045, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030, A386993.
%K A386994 nonn,new
%O A386994 0,5
%A A386994 _Omar E. Pol_, Aug 27 2025
%E A386994 More terms from _Alois P. Heinz_, Aug 27 2025