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 A384930 #86 Aug 28 2025 16:46:03
%S A384930 1,3,3,1,7,5,1,12,7,1,15,9,3,1,15,3,11,1,28,13,1,21,3,15,8,1,31,17,1,
%T A384930 39,19,1,42,21,7,3,1,33,3,23,1,60,25,5,1,39,3,27,9,3,1,56,29,1,72,31,
%U A384930 1,63,33,11,3,1,51,3,35,12,1,91,37,1,57,3,39,13,3,1,90,41,1,96,43,1,77,7,45,32,1
%N A384930 Irregular triangle read by rows: T(n,k) is the sum of the terms of the (n-k+1)-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.
%C A384930 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.
%C A384930 The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
%C A384930 At least for the first 1000 rows the row lengths coincide with A237271.
%C A384930 Note that if the conjectures related to the 2-dense sublists of divisors of n are true so we have that essentially all sequences where the words "part" or "parts" are mentioned having cf. A237593 are also related to the 2-dense sublists of divisors of n, for example the square array A240062.
%H A384930 Paolo Xausa, <a href="/A384930/b384930.txt">Table of n, a(n) for n = 1..12242</a> (rows 1..4000 of triangle, flattened).
%F A384930 T(n,k) = A384149(n,m+1-k), n >= 1, k >= 1, and m is the length of row n.
%F A384930 T(n,k) = 2*A237270(n,k) - A384149(n,k), n >= 1, k >= 1, (conjectured).
%e A384930 ---------------------------------------------------------------------
%e A384930 | n | Row n of | List of divisors of n | Number of |
%e A384930 | | the triangle | [with sublists in brackets] | sublists |
%e A384930 ---------------------------------------------------------------------
%e A384930 | 1 | 1; | [1]; | 1 |
%e A384930 | 2 | 3; | [1, 2]; | 1 |
%e A384930 | 3 | 3, 1; | [1], [3]; | 2 |
%e A384930 | 4 | 7; | [1, 2, 4]; | 1 |
%e A384930 | 5 | 5, 1; | [1], [5]; | 2 |
%e A384930 | 6 | 12; | [1, 2, 3, 6]; | 1 |
%e A384930 | 7 | 7, 1; | [1], [7]; | 2 |
%e A384930 | 8 | 15; | [1, 2, 4, 8]; | 1 |
%e A384930 | 9 | 9, 3, 1; | [1], [3], [9]; | 3 |
%e A384930 | 10 | 15 3; | [1, 2], [5, 10]; | 2 |
%e A384930 | 11 | 11, 1; | [1], [11]; | 2 |
%e A384930 | 12 | 28; | [1, 2, 3, 4, 6, 12]; | 1 |
%e A384930 | 13 | 13, 1; | [1], [13]; | 2 |
%e A384930 | 14 | 21, 3; | [1, 2], [7, 14]; | 2 |
%e A384930 | 15 | 15, 8, 1; | [1], [3, 5], [15]; | 3 |
%e A384930 | 16 | 31; | [1, 2, 4, 8, 16]; | 1 |
%e A384930 | 17 | 17, 1; | [1], [17]; | 2 |
%e A384930 | 18 | 39; | [1, 2, 3, 6, 9, 18]; | 1 |
%e A384930 | 19 | 19, 1; | [1], [19]; | 2 |
%e A384930 | 20 | 42; | [1, 2, 4, 5, 10, 20]; | 1 |
%e A384930 | 21 | 21, 7, 3, 1; | [1], [3], [7], [21]; | 4 |
%e A384930 | 22 | 33, 3; | [1, 2], [11, 22]; | 2 |
%e A384930 | 23 | 23, 1; | [1], [23]; | 2 |
%e A384930 | 24 | 60; | [1, 2, 3, 4, 6, 8, 12, 24]; | 1 |
%e A384930 ...
%e A384930 A conjectured relationship between a palindromic composition of sigma_0(n) = A000005(n) as n-th row of A384222 and the list of divisors of n as the n-th row of A027750 and a palindromic composition of sigma_1(n) = A000203(n) as the n-th row of A237270 and the diagram called "symmetric representation of sigma(n)" is as shown below with two examples.
%e A384930 .
%e A384930 For n = 10 the conjectured relationship is:
%e A384930 10th row of A384222.......................: [ 2 ], [ 2 ]
%e A384930 10th row of A027750.......................: 1, 2, 5, 10
%e A384930 10th row of A027750 with sublists.........: [ 1, 2 ], [ 5, 10]
%e A384930 10th row of A384149.......................: [ 3 ], [ 15 ]
%e A384930 10th row of this triangle.................: [ 15 ], [ 3 ]
%e A384930 10th row of the virtual sequence 2*A237270: [ 18 ], [ 18 ]
%e A384930 10th row of A237270.......................: [ 9 ], [ 9 ]
%e A384930 .
%e A384930 The symmetric representation of sigma_1(10) in the first quadrant is as follows:
%e A384930 .
%e A384930 _ _ _ _ _ _ 9
%e A384930 |_ _ _ _ _ |
%e A384930 | |_
%e A384930 |_ _|_
%e A384930 | |_ _ 9
%e A384930 |_ _ |
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 |_|
%e A384930 .
%e A384930 The diagram has two parts (or polygons) of areas [9], [9] respectively, so the 10th row of A237270 is [9], [9] and sigma_1(10) = A000203(10) = 18.
%e A384930 .
%e A384930 For n = 15 the conjectured relationship is:
%e A384930 15th row of A384222.......................: [ 1], [ 2 ], [ 1]
%e A384930 15th row of A027750.......................: 1, 3, 5, 15
%e A384930 15th row of A027750 with sublists.........: [ 1], [ 3, 5], [15]
%e A384930 15th row of A384149.......................: [ 1], [ 8 ], [15]
%e A384930 15th row of this triangle.................: [15], [ 8 ], [ 1]
%e A384930 15th row of the virtual sequence 2*A237270: [16], [ 16 ], [16]
%e A384930 15th row of A237270.......................: [ 8], [ 8 ], [ 8]
%e A384930 .
%e A384930 The symmetric representation of sigma_1(15) in the first quadrant is as follows:
%e A384930 .
%e A384930 _ _ _ _ _ _ _ _ 8
%e A384930 |_ _ _ _ _ _ _ _|
%e A384930 |
%e A384930 |_ _
%e A384930 |_ |_ 8
%e A384930 | |_
%e A384930 |_ _ |
%e A384930 |_|_ _ _ 8
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 | |
%e A384930 |_|
%e A384930 .
%e A384930 The diagram has three parts (or polygons) of areas [8], [8], [8] respectively, so the 15th row of A237270 is [8], [8], [8] and sigma_1(15) = A000203(15) = 24.
%e A384930 .
%e A384930 For the relationship with Dyck paths, partitions of n into consecutive parts and odd divisors of n see A237593, A235791, A237591 and A379630.
%t A384930 A384930row[n_] := Reverse[Total[Split[Divisors[n], #2 <= 2*# &], {2}]];
%t A384930 Array[A384930row, 50] (* _Paolo Xausa_, Aug 14 2025 *)
%Y A384930 Mirror of A384149.
%Y A384930 Row sums give A000203.
%Y A384930 Cf. A000105, A174973, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A240062, A245092, A262626, A379288, A379630, A384222, A384225, A384226, A384227.
%K A384930 nonn,tabf,changed
%O A384930 1,2
%A A384930 _Omar E. Pol_, Jul 19 2025