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%I A249351 #81 Aug 02 2023 14:33:32 %S A249351 1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1,1,1,1,1,1,2,1,1,1, %T A249351 1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, %U A249351 1,0,0,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1 %N A249351 Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n). %C A249351 Here T(n,k) is defined to be the "k-th width" of the symmetric representation of sigma(n), with n>=1 and 1<=k<=2n-1. Explanation: consider the diagram of the symmetric representation of sigma(n) described in A236104, A237593 and other related sequences. Imagine that the diagram for sigma(n) contains 2n-1 equidistant segments which are parallel to the main diagonal [(0,0),(n,n)] of the quadrant. The segments are located on the diagonal of the cells. The distance between two parallel segment is equal to sqrt(2)/2. T(n,k) is the length of the k-th segment divided by sqrt(2). Note that the triangle contains nonnegative terms because for some n the value of some widths is equal to zero. For an illustration of some widths see _Hartmut F. W. Hoft_'s contribution in the Links section of A237270. %C A249351 Row n has length 2*n-1. %C A249351 Row sums give A000203. %C A249351 If n is a power of 2 then all terms of row n are 1's. %C A249351 If n is an even perfect number then all terms of row n are 1's except the middle term which is 2. %C A249351 If n is an odd prime then row n lists (n+1)/2 1's, n-2 zeros, (n+1)/2 1's. %C A249351 The number of blocks of positive terms in row n gives A237271(n). %C A249351 The sum of the k-th block of positive terms in row n gives A237270(n,k). %C A249351 It appears that the middle diagonal is also A067742 (which was conjectured by _Michel Marcus_ in the entry A237593 and checked with two Mathematica functions up to n = 100000 by _Hartmut F. W. Hoft_). %C A249351 It appears that the trapezoidal numbers (A165513) are also the numbers k > 1 with the property that some of the noncentral widths of the symmetric representation of sigma(k) are not equal to 1. - _Omar E. Pol_, Mar 04 2023 %H A249351 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a> %e A249351 Triangle begins: %e A249351 1; %e A249351 1,1,1; %e A249351 1,1,0,1,1; %e A249351 1,1,1,1,1,1,1; %e A249351 1,1,1,0,0,0,1,1,1; %e A249351 1,1,1,1,1,2,1,1,1,1,1; %e A249351 1,1,1,1,0,0,0,0,0,1,1,1,1; %e A249351 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1; %e A249351 1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1; %e A249351 1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1; %e A249351 1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1; %e A249351 1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1; %e A249351 ... %e A249351 --------------------------------------------------------------------------- %e A249351 . Written as an isosceles triangle Diagram of %e A249351 . the sequence begins: the symmetry of sigma %e A249351 --------------------------------------------------------------------------- %e A249351 . _ _ _ _ _ _ _ _ _ _ _ _ %e A249351 . 1; |_| | | | | | | | | | | | %e A249351 . 1,1,1; |_ _|_| | | | | | | | | | %e A249351 . 1,1,0,1,1; |_ _| _|_| | | | | | | | %e A249351 . 1,1,1,1,1,1,1; |_ _ _| _|_| | | | | | %e A249351 . 1,1,1,0,0,0,1,1,1; |_ _ _| _| _ _|_| | | | %e A249351 . 1,1,1,1,1,2,1,1,1,1,1; |_ _ _ _| _| | _ _|_| | %e A249351 . 1,1,1,1,0,0,0,0,0,1,1,1,1; |_ _ _ _| |_ _|_| _ _| %e A249351 . 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1; |_ _ _ _ _| _| | %e A249351 . 1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1; |_ _ _ _ _| | _| %e A249351 . 1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _| _ _| %e A249351 . 1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1; |_ _ _ _ _ _| | %e A249351 .1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _ _| %e A249351 ... %e A249351 From _Omar E. Pol_, Nov 22 2020: (Start) %e A249351 Also consider the infinite double-staircases diagram defined in A335616. %e A249351 For n = 15 the diagram with first 15 levels looks like this: %e A249351 . %e A249351 Level "Double-staircases" diagram %e A249351 . _ %e A249351 1 _|1|_ %e A249351 2 _|1 _ 1|_ %e A249351 3 _|1 |1| 1|_ %e A249351 4 _|1 _| |_ 1|_ %e A249351 5 _|1 |1 _ 1| 1|_ %e A249351 6 _|1 _| |1| |_ 1|_ %e A249351 7 _|1 |1 | | 1| 1|_ %e A249351 8 _|1 _| _| |_ |_ 1|_ %e A249351 9 _|1 |1 |1 _ 1| 1| 1|_ %e A249351 10 _|1 _| | |1| | |_ 1|_ %e A249351 11 _|1 |1 _| | | |_ 1| 1|_ %e A249351 12 _|1 _| |1 | | 1| |_ 1|_ %e A249351 13 _|1 |1 | _| |_ | 1| 1|_ %e A249351 14 _|1 _| _| |1 _ 1| |_ |_ 1|_ %e A249351 15 |1 |1 |1 | |1| | 1| 1| 1| %e A249351 . %e A249351 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 A249351 . %e A249351 Level "Ziggurat" diagram %e A249351 . _ %e A249351 6 |1| %e A249351 7 _ | | _ %e A249351 8 _|1| _| |_ |1|_ %e A249351 9 _|1 | |1 1| | 1|_ %e A249351 10 _|1 | | | | 1|_ %e A249351 11 _|1 | _| |_ | 1|_ %e A249351 12 _|1 | |1 1| | 1|_ %e A249351 13 _|1 | | | | 1|_ %e A249351 14 _|1 | _| _ |_ | 1|_ %e A249351 15 |1 | |1 |1| 1| | 1| %e A249351 . %e A249351 The 15th row %e A249351 of this seq: [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 A249351 The 15th row %e A249351 of A237270: [ 8, 8, 8 ] %e A249351 The 15th row %e A249351 of A296508: [ 8, 7, 1, 0, 8 ] %e A249351 The 15th row %e A249351 of A280851 [ 8, 7, 1, 8 ] %e A249351 . %e A249351 The number of horizontal steps (or 1's) in the successive columns of the above diagram gives the 15th row of this triangle. %e A249351 For more information about the parts of the symmetric representation of sigma(n) see A237270. For more information about the subparts see A239387, A296508, A280851. %e A249351 More generally, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n. (End) %t A249351 (* function segments are defined in A237270 *) %t A249351 a249351[n_] := Flatten[Map[segments, Range[n]]] %t A249351 a249351[10] (* _Hartmut F. W. Hoft_, Jul 20 2022 *) %Y A249351 Cf. A000203, A003056, A067742, A071562, A165513, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A245092, A245685, A246955, A246956, A247687, A249223, A250068, A250070, A250071, A262626, A280850, A280851, A296508, A235616, A347186. %K A249351 nonn,tabf %O A249351 1,31 %A A249351 _Omar E. Pol_, Oct 26 2014