A390610 Irregular triangle read by rows: T(n,k) is the sum of composite numbers in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

0, 0, 0, 0, 4, 0, 0, 6, 0, 0, 12, 0, 0, 9, 0, 10, 0, 0, 22, 0, 0, 0, 14, 0, 0, 15, 28, 0, 0, 33, 0, 0, 34, 0, 0, 0, 21, 0, 22, 0, 0, 54, 0, 0, 25, 0, 26, 0, 0, 9, 27, 46, 0, 0, 61, 0, 0, 60, 0, 0, 0, 33, 0, 34, 0, 0, 35, 85, 0, 0, 0, 38, 0, 0, 0, 39, 82, 0, 0, 83, 0, 0, 4, 66, 0, 24, 45, 0, 46, 0, 0, 118, 0, 0, 49

Offset

1,5

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.

At least for the first 1000 rows the row lengths give A237271.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Formula

T(n,k) = A384149(n,k) - A390609(n,k).

Example

Triangle begins:
   0;
   0
   0,  0;
   4;
   0,  0;
   6;
   0,  0;
  12;
   0,  0,  9;
   0, 10;
   0,  0;
  22;
   0,  0;
   0, 14;
   0,  0, 15;
  28;
  ...
For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. In the first 2-dense sublist the sum of composite numbers is 6, so T(78,1) = 6. In the second 2-dense sublist the sum of composite numbers is 26 + 39 + 78 = 143, so T(78,2) = 143.

Mathematica

A390610row[n_] := Map[Total[Select[#, CompositeQ]] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A390610row, 50] (* Paolo Xausa, Nov 18 2025 *)

Crossrefs

Row sums give A023891.

Cf. A002808, A027750, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A386604, A386989, A386992, A387030, A389592, A389761, A389762, A390443, A390494, A390495, A390609.

Sequence in context: A212044 A290335 A341795 * A063730 A355945 A386403

Adjacent sequences: A390607 A390608 A390609 * A390611 A390612 A390613

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 12 2025

Status

approved