A394054 - OEIS

%I #13 Mar 18 2026 22:52:26

%S 1,2,2,1,3,1,3,2,4,2,4,3,5,3,5,3,1,5,4,1,5,5,1,6,5,1,6,6,1,6,7,1,6,7,

%T 2,7,7,2,7,8,2,8,8,2,8,9,2,9,9,2,9,9,2,1,9,10,2,1,9,11,2,1,10,11,2,1,

%U 10,11,3,1,10,12,3,1,10,12,3,2,11,12,3,2,11,13,3,2,12,13,3,2,12,14,3,2

%N Irregular triangle read by rows: T(n,k) is the number of positive integers <= n with exactly k 2-dense sublists of divisors.

%C For the definition of 2-dense sublists of divisors see A384222.

%C Observation: at least for the first 10000 rows the row numbers where the triangle widens are given by A239663.

%C Conjecture: T(n,k) is the number of positive integers m <= n whose symmetric representation of sigma(m) has k parts.

%H Paolo Xausa, <a href="/A394054/b394054.txt">Table of n, a(n) for n = 1..10504</a> (rows 1..1500 of triangle, flattened).

%e Triangle begins:

%e   -----------------

%e    n\k  1  2  3  4

%e   -----------------

%e    1 |  1;

%e    2 |  2;

%e    3 |  2, 1;

%e    4 |  3, 1;

%e    5 |  3, 2;

%e    6 |  4, 2;

%e    7 |  4, 3;

%e    8 |  5, 3;

%e    9 |  5, 3, 1;

%e   10 |  5, 4, 1;

%e   11 |  5, 5, 1;

%e   12 |  6, 5, 1;

%e   13 |  6, 6, 1;

%e   14 |  6, 7, 1;

%e   15 |  6, 7, 2;

%e   16 |  7, 7, 2;

%e   17 |  7, 8, 2;

%e   18 |  8, 8, 2;

%e   19 |  8, 9, 2;

%e   20 |  9, 9, 2;

%e   21 |  9, 9, 2, 1;

%e   ...

%Y Row sums give A000027.

%Y Column 1 gives A392987.

%Y Analogous to A394052 and A394053.

%Y Cf. A174973 (2-dense numbers), A237270, A237271, A237593, A239663, A240062, A379288, A384149, A384222.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Mar 12 2026