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A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

%I A386989 #20 Aug 29 2025 11:08:50
%S A386989 1,2,1,3,8,1,5,36,1,7,64,1,3,9,2,50,1,11,1728,1,13,2,98,1,15,15,1024,
%T A386989 1,17,5832,1,19,8000,1,3,7,21,2,242,1,23,331776,1,5,25,2,338,1,3,9,27,
%U A386989 21952,1,29,810000,1,31,32768,1,3,11,33,2,578,1,35,35,10077696,1,37,2,722,1,3,13,39,2560000
%N A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.
%C A386989 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 A386989 The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
%C A386989 It is conjectured that row lengths are given by A237271.
%H A386989 Paolo Xausa, <a href="/A386989/b386989.txt">Table of n, a(n) for n = 1..10607</a> (rows 1..3500 of triangle, flattened).
%e A386989 Triangle begins:
%e A386989    1;
%e A386989    2;
%e A386989    1,  3;
%e A386989    8;
%e A386989    1,  5;
%e A386989   36;
%e A386989    1,  7;
%e A386989   64;
%e A386989    1,  3,  9;
%e A386989    2, 50;
%e A386989   ...
%e A386989 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]. The product of terms are 1*2 = 2 and 5*10 = 50 respectively, so the row 10 of the triangle is [2, 50].
%t A386989 A386989row[n_] :=Times @@@ Split[Divisors[n], #2/# <= 2 &];
%t A386989 Array[A386989row, 50] (* _Paolo Xausa_, Aug 29 2025 *)
%Y A386989 Row products give A007955.
%Y A386989 Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984.
%K A386989 nonn,tabf,changed
%O A386989 1,2
%A A386989 _Omar E. Pol_, Aug 12 2025