A394054 Irregular triangle read by rows: T(n,k) is the number of positive integers <= n with exactly k 2-dense sublists of divisors.

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, 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, 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

Offset

1,2

Comments

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

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

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..10504 (rows 1..1500 of triangle, flattened).

Example

Triangle begins:
  -----------------
   n\k  1  2  3  4
  -----------------
   1 |  1;
   2 |  2;
   3 |  2, 1;
   4 |  3, 1;
   5 |  3, 2;
   6 |  4, 2;
   7 |  4, 3;
   8 |  5, 3;
   9 |  5, 3, 1;
  10 |  5, 4, 1;
  11 |  5, 5, 1;
  12 |  6, 5, 1;
  13 |  6, 6, 1;
  14 |  6, 7, 1;
  15 |  6, 7, 2;
  16 |  7, 7, 2;
  17 |  7, 8, 2;
  18 |  8, 8, 2;
  19 |  8, 9, 2;
  20 |  9, 9, 2;
  21 |  9, 9, 2, 1;
  ...

Crossrefs

Row sums give A000027.

Column 1 gives A392987.

Analogous to A394052 and A394053.

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

Sequence in context: A237591 A359979 A277730 * A174167 A374064 A159876

Adjacent sequences: A394051 A394052 A394053 * A394055 A394056 A394057

Keyword

nonn,tabf

Author

Omar E. Pol, Mar 12 2026

Status

approved