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

1,3

Comments

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.

Links

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

Formula

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A334144 A232551 A367006 * A261129 A309121 A305871

Adjacent sequences: A386990 A386991 A386992 * A386994 A386995 A386996

Keyword

nonn

Author

Omar E. Pol, Aug 23 2025

Status

approved