A387030 Irregular triangle read by rows: T(n,k) is the number of primes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 3, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Author

Omar E. Pol, Aug 13 2025

Keywords

• nonn
• tabf
• changed

Comments

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.

Examples

			Triangle begins:
  0;
  1;
  0, 1;
  1;
  0, 1;
  2;
  0, 1;
  1;
  0, 1, 0;
  1, 1;
  0, 1;
  2;
  0, 1;
  1, 1;
  0, 2, 0;
  ...
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 prime 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 second sublist contains primes, so row 15 is [0, 2, 0].
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12242 (rows 1..4000 of triangle, flattened).

Crossrefs

Row sums give A001221.

Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930.

Programs

Mathematica

A387030row[n_] := Map[Count[#, _?PrimeQ] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A387030row, 50] (* Paolo Xausa, Aug 19 2025 *)