A386992 -id:A386992 - cp's OEIS frontend

A386993 Number of 2-dense sublists of divisors of the n-th squarefree number.

1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 1, 2, 4, 2, 3, 2, 2, 4, 2, 1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 4, 1, 2, 4, 3, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 4, 2, 3, 4, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 3, 4, 2, 2, 4, 2, 4, 2, 4, 2, 4, 3, 2, 4, 2, 2, 2, 2, 4, 2, 3, 4, 2, 2, 2, 3, 4, 2, 2, 4, 4, 2, 5, 2, 2, 3, 2

Offset: 1

Omar E. Pol, Aug 23 2025

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.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

			For n = 11 the 11th squarefree 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(11) = 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A005117, A174973 (2-dense numbers), A237271, A379288, A380328, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030.

Map[Length[Split[Divisors[#], #2 <= 2*# &]] &, Select[Range[150], SquareFreeQ]] (* Paolo Xausa, Aug 29 2025 *)

a(n) = A237271(A005117(n)). (conjectured).

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

Original entry on oeis.org

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, 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, 17, 8, 7, 8, 20, 9, 15, 4, 1, 4, 16, 18, 24, 15, 7, 4, 3, 5

Offset: 0

Omar E. Pol, Aug 27 2025

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.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

			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.

Cf. A000045, A133021, A139045, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030, A386993.

A386994[n_] := Length[Split[Divisors[Fibonacci[n]], #2 <= 2*# &]];
Array[A386994, 100, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = A237271(A000045(n)), n >= 1. (conjectured).

More terms from Alois P. Heinz, Aug 27 2025

cp's OEIS Frontend

A386993 Number of 2-dense sublists of divisors of the n-th squarefree number.

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