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.
The 10x8 section of the table T(w,p) with dashes indicating values greater than 120*10^6; rows w denote the common maximum width in all parts and columns p the number of parts in the symmetric representation of sigma(T(w,p)). w\p | 1 2 3 4 5 6 7 8 ... -------------------------------------------------------------------------- 1 | 1 3 9 21 81 147 729 903 2 | 6 78 1014 12246 171366 1922622 28960854 - 3 | 60 7620 967740 116136420 - - - 4 | 120 28920 6969720 - 5 | 360 261720 - 6 | 840 1422120 - 7 | 3360 22622880 - 8 | 2520 12728520 - 9 | 5040 50858640 - 10| 10080 - ... The symmetric representation of sigma for T(2,2) = 78 consists of the two parts (84, 84) of maximum widths (2, 2), and that of T(2,3) = 1014 consists of the three parts (1020, 156, 1020) of maximum widths (2, 2, 2).
(* function a341969 is defined in A341969 *) a348142[n_, {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, Monitor[For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], #!=0&], #[[1]]!=0&]]; c=Length[s]; u=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]], list]; list] table=a348142[120000000, {10, 10}] (* 10x10 table; very long computation time *) p[n_] := n-row[n-1](row[n-1]+1)/2 w[n_] := row[n-1]-p[n]+2 Map[table[[w[#], p[#]]]&, Range[23]] (* sequence data *)
Sequence values for the first 4 powers of 3: {a(1), a(3), a(9), a(27)} = {1, 2, 5, 10} = {1, 10, 101, 1010}.
Table for a(1..16), a(27) and a(28) together with their lists of the base-2 representation, of the odd/even positions of 1's in the n-th row of A237048, and of the sizes of the parts in SRS(n):
n a(n) odd/even A237048 A237270
1 1 {1} {1} {1}
2 1 {1} {1} {3}
3 2 {1,0} {1,1} {2,2}
4 1 {1} {1,0} {7}
5 2 {1,0} {1,1} {3,3}
6 3 {1,1} {1,0,1} {12}
7 2 {1,0} {1,1,0} {4,4}
8 1 {1} {1,0,0} {15}
9 5 {1,0,1} {1,1,1} {5,3,5}
10 2 {1,0} {1,0,0,1} {9,9}
11 2 {1,0} {1,1,0,0} {6,6}
12 3 {1,1} {1,0,1,0} {28}
13 2 {1,0} {1,1,0,0} {7,7}
14 2 {1,0} {1,0,0,1} {12,12}
15 11 {1,0,1,1} {1,1,1,0,1} {8,8,8}
16 1 {1} {1,0,0,0,0} {31}
...
27 10 {1,0,1,0} {1,1,1,0,0,1} {14,6,6,14}
28 3 {1,1} {1,0,0,0,0,0,1} {56}
...
(* function a237048[ ] is defined in A237048 *) b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]] a352696[n_] := Map[b237048, Range[n]] a352696[85]
357 is in the sequence because the 357th row of A237593 is [179, 60, 31, 18, 12, 9, 7, 6, 4, 4, 3, 3, 2, 3, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 4, 4, 6, 7, 9, 12, 18, 31, 60, 179], and the 356th row of the same triangle is [179, 60, 30, 18, 13, 9, 6, 6, 4, 4, 3, 3, 3, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 3, 3, 3, 4, 4, 6, 6, 9, 13, 18, 30, 60, 179], therefore between both symmetric Dyck paths there are seven parts: [179, 61, 29, 38, 29, 61, 179]. Note that the sum of these parts is 179 + 61 + 29 + 38 + 29 + 61 + 179 = 576, equaling the sum of the divisors of 357: 1 + 3 + 7 + 17 + 21 + 51 + 119 + 357 = 576. (The diagram of the symmetric representation of sigma(357) = 576 is too large to include.)
Triangle begins:
4;
6;
4, 4;
10;
4, 4;
12;
4, 4;
14;
4, 6, 4;
8, 8;
4, 4;
18;
4, 4;
8, 8;
4, 12, 4;
...
Illustration of row 9:
4
_ _ _ _ _
|_ _ _ _ _|
|_ _ 6
|_ |
|_|_ _
| |
| |
| | 4
| |
|_|
.
For n = 9 the symmetric representation of sigma(9) has three parts from left to right as follows: a rectangle, a concave hexagon and a rectangle. The number of edges of the polygons are 4, 6, 4 respectively, so the row 9 of the triangle is [4, 6, 4].
Illustration of a(9) = 14:
4
_ _ _ _ _
|_ _ _ _ _|
|_ _ 6
|_ |
|_|_ _
| |
| |
| | 4
| |
|_|
.
For n = 9 the symmetric representation of sigma(9) has three parts from right to left as follows: a rectangle, a concave hexagon and a rectangle. The number of edges of the polygons are 4, 6, 4 respectively, therefore the total number of edges is 4 + 6 + 4 = 14, so a(9) = 14.
a(5) = 128 = 2^7 has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1. a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal. A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1): --------------------- 2i+1 /---- p ----/ --------------------- 1 3 .. 3 3 5 .. 13 5 11 .. 61 7 17 .. 251 9 37 .. 1021 ...
(* a2[ ] and its support functions are defined in A249223 *) a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]] a365265[{m_, n_}] := Select[Range[m, n], a365265Q] a365265[{1,75000}]
Comments