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 A237270 #188 Jun 01 2025 09:58:46
%S A237270 1,3,2,2,7,3,3,12,4,4,15,5,3,5,9,9,6,6,28,7,7,12,12,8,8,8,31,9,9,39,
%T A237270 10,10,42,11,5,5,11,18,18,12,12,60,13,5,13,21,21,14,6,6,14,56,15,15,
%U A237270 72,16,16,63,17,7,7,17,27,27,18,12,18,91,19,19,30,30,20,8,8,20,90
%N A237270 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n).
%C A237270 T(n,k) is the number of cells in the k-th region of the n-th set of regions in a diagram of the symmetry of sigma(n), see example.
%C A237270 Row n is a palindromic composition of sigma(n).
%C A237270 Row sums give A000203.
%C A237270 Row n has length A237271(n).
%C A237270 In the row 2n-1 of triangle both the first term and the last term are equal to n.
%C A237270 If n is an odd prime then row n is [m, m], where m = (1 + n)/2.
%C A237270 The connection with A196020 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A239660 --> this sequence.
%C A237270 For the boundary segments in an octant see A237591.
%C A237270 For the boundary segments in a quadrant see A237593.
%C A237270 For the boundary segments in the spiral see also A239660.
%C A237270 For the parts in every quadrant of the spiral see A239931, A239932, A239933, A239934.
%C A237270 We can find the spiral on the terraces of the stepped pyramid described in A244050. - _Omar E. Pol_, Dec 07 2016
%C A237270 T(n,k) is also the area of the k-th terrace, from left to right, at the n-th level, starting from the top, of the stepped pyramid described in A245092 (see Links section). - _Omar E. Pol_, Aug 14 2018
%H A237270 Robert Price, <a href="/A237270/b237270.txt">Table of n, a(n) for n = 1..15542</a> (rows n = 1..5000, flattened)
%H A237270 Hartmut F. W. Hoft, <a href="/A237270/a237270_1.pdf">Sample visual documentation for Mathematica code</a>
%H A237270 Michel Marcus, <a href="/A244145/a244145_3.pdf">A colored version of the symmetric representation of sigma(n), multipage, n = 1..85</a>
%H A237270 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid (A237593, A237270, A262626)</a>
%H A237270 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (16 levels)</a>
%H A237270 Omar E. Pol, <a href="/A237270/a237270.jpg">Perspective view of the stepped pyramid into four quadrants (11 levels)</a>. This is formed by combing four copies of the pyramid back-to-back (cf. A244050).
%H A237270 N. J. A. Sloane, <a href="/A237270/a237270_2.pdf">Another drawing of the spiral</a>
%H A237270 N. J. A. Sloane, <a href="/A237270/a237270_3.pdf">Spiral showing only the outer boundary</a>
%H A237270 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A237270 T(n, k) = (A384149(n, k) + A384149(n, m+1-k))/2, where m = A237271(n) is the row length. (conjectured) - _Peter Munn_, Jun 01 2025
%e A237270 Illustration of the first 27 terms as regions (or parts) of a spiral constructed with the first 15.5 rows of A239660:
%e A237270 .
%e A237270 . _ _ _ _ _ _ _ _
%e A237270 . | _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
%e A237270 . | | |_ _ _ _ _ _ _|
%e A237270 . 12 _| | |
%e A237270 . |_ _| _ _ _ _ _ _ |_ _
%e A237270 . 12 _ _| | _ _ _ _ _|_ _ _ _ _ 5 |_
%e A237270 . _ _ _| | 9 _| | |_ _ _ _ _| |
%e A237270 . | _ _ _| 9 _|_ _| |_ _ 3 |_ _ _ 7
%e A237270 . | | _ _| | _ _ _ _ |_ | | |
%e A237270 . | | | _ _| 12 _| _ _ _|_ _ _ 3 |_|_ _ 5 | |
%e A237270 . | | | | _| | |_ _ _| | | | |
%e A237270 . | | | | | _ _| |_ _ 3 | | | |
%e A237270 . | | | | | | 3 _ _ | | | | | |
%e A237270 . | | | | | | | _|_ 1 | | | | | |
%e A237270 . _|_| _|_| _|_| _|_| |_| _|_| _|_| _|_| _
%e A237270 . | | | | | | | | | | | | | | | |
%e A237270 . | | | | | | |_|_ _ _| | | | | | | |
%e A237270 . | | | | | | 2 |_ _|_ _| _| | | | | | |
%e A237270 . | | | | |_|_ 2 |_ _ _|7 _ _| | | | | |
%e A237270 . | | | | 4 |_ _| _ _| | | | |
%e A237270 . | | |_|_ _ |_ _ _ _ | _| _ _ _| | | |
%e A237270 . | | 6 |_ |_ _ _ _|_ _ _ _| | 15 _| _ _| | |
%e A237270 . |_|_ _ _ |_ 4 |_ _ _ _ _| _| | _ _ _| |
%e A237270 . 8 | |_ _ | | _| | _ _ _|
%e A237270 . |_ | |_ _ _ _ _ _ | _ _|28 _| |
%e A237270 . |_ |_ |_ _ _ _ _ _|_ _ _ _ _ _| | _| _|
%e A237270 . 8 |_ _| 6 |_ _ _ _ _ _ _| _ _| _|
%e A237270 . | | _ _| 31
%e A237270 . |_ _ _ _ _ _ _ _ | |
%e A237270 . |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
%e A237270 . 8 |_ _ _ _ _ _ _ _ _|
%e A237270 .
%e A237270 .
%e A237270 [For two other drawings of the spiral see the links. - _N. J. A. Sloane_, Nov 16 2020]
%e A237270 If the sequence does not contain negative terms then its terms can be represented in a quadrant. For the construction of the diagram we use the symmetric Dyck paths of A237593 as shown below:
%e A237270 ---------------------------------------------------------------
%e A237270 Triangle Diagram of the symmetry of sigma (n = 1..24)
%e A237270 ---------------------------------------------------------------
%e A237270 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A237270 1; |_| | | | | | | | | | | | | | | | | | | | | | | |
%e A237270 3; |_ _|_| | | | | | | | | | | | | | | | | | | | | |
%e A237270 2, 2; |_ _| _|_| | | | | | | | | | | | | | | | | | | |
%e A237270 7; |_ _ _| _|_| | | | | | | | | | | | | | | | | |
%e A237270 3, 3; |_ _ _| _| _ _|_| | | | | | | | | | | | | | | |
%e A237270 12; |_ _ _ _| _| | _ _|_| | | | | | | | | | | | | |
%e A237270 4, 4; |_ _ _ _| |_ _|_| _ _|_| | | | | | | | | | | |
%e A237270 15; |_ _ _ _ _| _| | _ _ _|_| | | | | | | | | |
%e A237270 5, 3, 5; |_ _ _ _ _| | _|_| | _ _ _|_| | | | | | | |
%e A237270 9, 9; |_ _ _ _ _ _| _ _| _| | _ _ _|_| | | | | |
%e A237270 6, 6; |_ _ _ _ _ _| | _| _| _| | _ _ _ _|_| | | |
%e A237270 28; |_ _ _ _ _ _ _| |_ _| _| _ _| | | _ _ _ _|_| |
%e A237270 7, 7; |_ _ _ _ _ _ _| | _ _| _| _| | | _ _ _ _|
%e A237270 12, 12; |_ _ _ _ _ _ _ _| | | | _|_| |* * * *
%e A237270 8, 8, 8; |_ _ _ _ _ _ _ _| | _ _| _ _|_| |* * * *
%e A237270 31; |_ _ _ _ _ _ _ _ _| | _ _| _| _ _|* * * *
%e A237270 9, 9; |_ _ _ _ _ _ _ _ _| | |_ _ _| _|* * * * * *
%e A237270 39; |_ _ _ _ _ _ _ _ _ _| | _ _| _|* * * * * * *
%e A237270 10, 10; |_ _ _ _ _ _ _ _ _ _| | | |* * * * * * * *
%e A237270 42; |_ _ _ _ _ _ _ _ _ _ _| | _ _ _|* * * * * * * *
%e A237270 11, 5, 5, 11; |_ _ _ _ _ _ _ _ _ _ _| | |* * * * * * * * * * *
%e A237270 18, 18; |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *
%e A237270 12, 12; |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *
%e A237270 60; |_ _ _ _ _ _ _ _ _ _ _ _ _|* * * * * * * * * * *
%e A237270 ...
%e A237270 The total number of cells in the first n set of symmetric regions of the diagram equals A024916(n), the sum of all divisors of all positive integers <= n, hence the total number of cells in the n-th set of symmetric regions of the diagram equals sigma(n) = A000203(n).
%e A237270 For n = 9 the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 9 is [5, 3, 5].
%e A237270 The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.
%e A237270 For n = 24 the 24th row of A237593 is [13, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 13] and the 23rd row of A237593 is [12, 5, 2, 2, 1, 1, 1, 1, 2, 2, 5, 12] therefore between both symmetric Dyck paths there are only one region (or part) of size 60, so row 24 is 60.
%e A237270 The sum of divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = A000203(24) = 60. On the other hand the sum of the parts of the symmetric representation of sigma(24) is 60, equaling the sum of divisors of 24.
%e A237270 Note that the number of *'s in the diagram is 24^2 - A024916(24) = 576 - 491 = A004125(24) = 85.
%e A237270 From _Omar E. Pol_, Nov 22 2020: (Start)
%e A237270 Also consider the infinite double-staircases diagram defined in A335616 (see the theorem).
%e A237270 For n = 15 the diagram with first 15 levels looks like this:
%e A237270 .
%e A237270 Level "Double-staircases" diagram
%e A237270 . _
%e A237270 1 _|1|_
%e A237270 2 _|1 _ 1|_
%e A237270 3 _|1 |1| 1|_
%e A237270 4 _|1 _| |_ 1|_
%e A237270 5 _|1 |1 _ 1| 1|_
%e A237270 6 _|1 _| |1| |_ 1|_
%e A237270 7 _|1 |1 | | 1| 1|_
%e A237270 8 _|1 _| _| |_ |_ 1|_
%e A237270 9 _|1 |1 |1 _ 1| 1| 1|_
%e A237270 10 _|1 _| | |1| | |_ 1|_
%e A237270 11 _|1 |1 _| | | |_ 1| 1|_
%e A237270 12 _|1 _| |1 | | 1| |_ 1|_
%e A237270 13 _|1 |1 | _| |_ | 1| 1|_
%e A237270 14 _|1 _| _| |1 _ 1| |_ |_ 1|_
%e A237270 15 |1 |1 |1 | |1| | 1| 1| 1|
%e A237270 .
%e A237270 Starting from A196020 and after the algorithm described in A280850 and A296508 applied to the above diagram we have a new diagram as shown below:
%e A237270 .
%e A237270 Level "Ziggurat" diagram
%e A237270 . _
%e A237270 6 |1|
%e A237270 7 _ | | _
%e A237270 8 _|1| _| |_ |1|_
%e A237270 9 _|1 | |1 1| | 1|_
%e A237270 10 _|1 | | | | 1|_
%e A237270 11 _|1 | _| |_ | 1|_
%e A237270 12 _|1 | |1 1| | 1|_
%e A237270 13 _|1 | | | | 1|_
%e A237270 14 _|1 | _| _ |_ | 1|_
%e A237270 15 |1 | |1 |1| 1| | 1|
%e A237270 .
%e A237270 The 15th row
%e A237270 of A249351 : [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
%e A237270 The 15th row
%e A237270 of triangle: [ 8, 8, 8 ]
%e A237270 The 15th row
%e A237270 of A296508: [ 8, 7, 1, 0, 8 ]
%e A237270 The 15th row
%e A237270 of A280851 [ 8, 7, 1, 8 ]
%e A237270 .
%e A237270 More generally, for n >= 1, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n.
%e A237270 For the definition of subparts see A239387 and also A296508, A280851. (End)
%t A237270 T[n_,k_] := Ceiling[(n + 1)/k - (k + 1)/2] (* from A235791 *)
%t A237270 path[n_] := Module[{c = Floor[(Sqrt[8n + 1] - 1)/2], h, r, d, rd, k, p = {{0, n}}}, h = Map[T[n, #] - T[n, # + 1] &, Range[c]]; r = Join[h, Reverse[h]]; d = Flatten[Table[{{1, 0}, {0, -1}}, {c}], 1];
%t A237270 rd = Transpose[{r, d}]; For[k = 1, k <= 2c, k++, p = Join[p, Map[Last[p] + rd[[k, 2]] * # &, Range[rd[[k, 1]]]]]]; p]
%t A237270 segments[n_] := SplitBy[Map[Min, Drop[Drop[path[n], 1], -1] - path[n - 1]], # == 0 &]
%t A237270 a237270[n_] := Select[Map[Apply[Plus, #] &, segments[n]], # != 0 &]
%t A237270 Flatten[Map[a237270, Range[40]]] (* data *)
%t A237270 (* _Hartmut F. W. Hoft_, Jun 23 2014 *)
%Y A237270 Cf. A000203, A004125, A023196, A024916, A153485, A196020, A221529, A231347, A235791, A235796, A236104, A236112, A236540, A237046, A237048, A237271, A237590, A237591, A237593, A239050, A239660, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A249351, A262626, A280850, A280851, A296508, A335616, A340035, A384149.
%K A237270 nonn,tabf,look
%O A237270 1,2
%A A237270 _Omar E. Pol_, Feb 19 2014
%E A237270 Drawing of the spiral extended by _Omar E. Pol_, Nov 22 2020