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.

%I A386992 #20 Aug 29 2025 01:57:42
%S A386992 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,
%T A386992 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,
%U A386992 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
%N 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.
%C A386992 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.
%C A386992 The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
%C A386992 It is conjectured that row lengths are given by A237271.
%H A386992 Paolo Xausa, <a href="/A386992/b386992.txt">Table of n, a(n) for n = 1..10607</a> (rows 1..3500 of triangle, flattened).
%F A386992 T(n,k) = A384222(n,k) - A387030(n,k).
%e A386992 Triangle begins:
%e A386992   1;
%e A386992   1;
%e A386992   1, 0;
%e A386992   2;
%e A386992   1, 0;
%e A386992   2;
%e A386992   1, 0;
%e A386992   3;
%e A386992   1, 0, 1;
%e A386992   1, 1;
%e A386992   1, 0;
%e A386992   4;
%e A386992   1, 0;
%e A386992   1, 1;
%e A386992   1, 0, 1;
%e A386992   ...
%e A386992 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].
%e A386992 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].
%t A386992 A386992row[n_] := Map[Count[#, _?(!PrimeQ[#] &)] &, Split[Divisors[n], #2 <= 2*# &]];
%t A386992 Array[A386992row, 50] (* _Paolo Xausa_, Aug 28 2025 *)
%Y A386992 Row sums give A033273.
%Y A386992 Cf. A018252, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386993, A387030.
%K A386992 nonn,tabf,new
%O A386992 1,5
%A A386992 _Omar E. Pol_, Aug 23 2025