This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384149 #33 Jun 15 2025 17:43:32
%S A384149 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,
%T A384149 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,
%U A384149 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
%N A384149 Irregular triangle T(n, k) in which row n gives the 2-densely-aggregated composition of sigma(n).
%C A384149 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.
%C A384149 We follow here the use of 2-dense/2-densely reported in the comments of A174973.
%C A384149 If n is in A174973 then row n has length 1 and its sole term is sigma(n).
%C A384149 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.
%e A384149 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).
%e A384149 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).
%e A384149 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).
%e A384149 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.
%e A384149 Triangle begins:
%e A384149 row
%e A384149 1 1,
%e A384149 2 3,
%e A384149 3 1, 3,
%e A384149 4 7,
%e A384149 5 1, 5,
%e A384149 6 12,
%e A384149 7 1, 7,
%e A384149 8 15,
%e A384149 9 1, 3, 9,
%e A384149 10 3, 15,
%e A384149 11 1, 11,
%e A384149 12 28,
%e A384149 13 1, 13,
%e A384149 14 3, 21,
%e A384149 15 1, 8, 15,
%e A384149 16 31,
%e A384149 ...
%e A384149 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.
%t A384149 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 *)
%t A384149 a384149[n_] := Flatten[Map[t384149, Range[n]]]
%t A384149 a384149[45] (* _Hartmut F. W. Hoft_, Jun 07 2025 *)
%Y A384149 Cf. A000203, A027750, A174973, A237270, A237271.
%K A384149 nonn,easy,tabf
%O A384149 1,2
%A A384149 _Peter Munn_, May 22 2025