A386993 -id:A386993 - cp's OEIS frontend

A386992 Irregular triangle read by rows: T(n,k) is the number of nonprimes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 3, 1, 0, 1, 1, 1, 1, 0, 4, 1, 0, 1, 1, 1, 0, 1, 4, 1, 0, 4, 1, 0, 4, 1, 0, 0, 1, 1, 1, 1, 0, 6, 1, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 0, 5, 1, 0, 5, 1, 0, 0, 1, 1, 1, 1, 0, 1, 7, 1, 0, 1, 1, 1, 0, 0, 1, 6, 1, 0, 5, 1, 0, 2, 2, 1, 2, 1, 1, 1, 1, 0, 8, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0

Offset: 1

Omar E. Pol, Aug 23 2025

In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
It is conjectured that row lengths are given by A237271.

			Triangle begins:
  1;
  1;
  1, 0;
  2;
  1, 0;
  2;
  1, 0;
  3;
  1, 0, 1;
  1, 1;
  1, 0;
  4;
  1, 0;
  1, 1;
  1, 0, 1;
  ...
For n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2] and [5, 10]. There is a nonprime number in each sublist, so row 10 is [1, 1].
For n = 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]. Only the first and the third sublists contain nonprimes, so row 15 is [1, 0, 1].

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Row sums give A033273.

Cf. A018252, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386993, A387030.

A386992row[n_] := Map[Count[#, _?(!PrimeQ[#] &)] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A386992row, 50] (* Paolo Xausa, Aug 28 2025 *)

T(n,k) = A384222(n,k) - A387030(n,k).

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

A386992 Irregular triangle read by rows: T(n,k) is the number of nonprimes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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