A384149 Irregular triangle T(n, k) in which row n gives the 2-densely-aggregated composition of sigma(n).

1, 3, 1, 3, 7, 1, 5, 12, 1, 7, 15, 1, 3, 9, 3, 15, 1, 11, 28, 1, 13, 3, 21, 1, 8, 15, 31, 1, 17, 39, 1, 19, 42, 1, 3, 7, 21, 3, 33, 1, 23, 60, 1, 5, 25, 3, 39, 1, 3, 9, 27, 56, 1, 29, 72, 1, 31, 63, 1, 3, 11, 33, 3, 51, 1, 12, 35, 91, 1, 37, 3, 57, 1, 3, 13, 39, 90, 1, 41, 96, 1, 43, 7, 77, 1, 32, 45

Offset: 1

Author

Peter Munn, May 22 2025

Keywords

• nonn
• easy
• tabf

Comments

We form the 2-densely-aggregated composition of sigma(n) = A000203(n) by listing the divisors of n in increasing order and assigning adjacent divisors for summation in the same aggregate if (and only if) they differ by a factor of less than or equal to 2. The ordering of the aggregate sums in the composition follows the ordering of the summed divisors.

We follow here the use of 2-dense/2-densely reported in the comments of A174973.

If n is in A174973 then row n has length 1 and its sole term is sigma(n).

From empirical evidence of the first 1000 rows, reinforced by some known related properties, it looks readily credible that if we take the average of row n and its reversal we get A237270 row n, a palindromic composition of sigma(n) that was defined using Dyck paths. Row lengths are therefore conjectured to be in the database as A237271.

Examples

			For row 9: the ordered divisors of 9 are (1, 3, 9). Adjacent divisors differ by a factor of 3, which is greater than 2, so each divisor is trivially summed into a separate aggregate and the 2-densely-aggregated composition of sigma(9) is (1, 3, 9).
For row 12, the ordered divisors of 12 are (1, 2, 3, 4, 6, 12). Every pair of adjacent divisors differs by a factor <= 2, so they are summed in a single aggregate and the 2-densely-aggregated composition of sigma(12) is (1+2+3+4+6+12) = (28).
For row 10, the ordered divisors of 10 are (1, 2, 5, 10). The adjacent divisors (1, 2) and (5, 10) differ by a factor of 2, but (2, 5) differ by a larger factor, so there are 2 aggregates and the 2-densely-aggregated composition of sigma(10) is (1+2, 5+10) = (3, 15).
For 29029 = 7 * 11 * 13 * 29, the 2-densely-aggregated composition of sigma(29029) is (1, 7+11+13, 29, 77+91+143+203+319+377, 1001, 2233+2639+4147, 29029) = (1, 31, 29, 1210, 1001, 9019, 29029). Note that this composition is not in ascending order.
Triangle begins:
  row
   1  1,
   2  3,
   3  1, 3,
   4  7,
   5  1, 5,
   6  12,
   7  1, 7,
   8  15,
   9  1, 3, 9,
  10  3, 15,
  11  1, 11,
  12  28,
  13  1, 13,
  14  3, 21,
  15  1, 8, 15,
  16  31,
  ...
If we take the average of row 9, (1, 3, 9) and its reversal, (9, 3, 1), we get (5, 3, 5), which is A237270 row 9. Doing the same for row 10, (3, 15), we get (9, 9), which is A237270 row 10.
		

Crossrefs

Cf. A000203, A027750, A174973, A237270, A237271.

Programs

Mathematica

t384149[n_] := Module[{dL = Divisors[n]}, Map[#[[1]] &, Map[Apply[Plus, #] &, Split[Transpose[{dL, Append[Rest[dL], 2 n + 1]}], #[[2]] <= 2 #[[1]] &]]]] (* row n of triangle *)
a384149[n_] := Flatten[Map[t384149, Range[n]]]
a384149[45] (* Hartmut F. W. Hoft, Jun 07 2025 *)