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A362817 Irregular triangle read by rows: T(n,k) (n>=1, k>=1) is the number of edges of the k-th polygon (or part), from left to right, of the symmetric representation of sigma(n).

%I A362817 #41 Aug 02 2023 14:34:08
%S A362817 4,6,4,4,10,4,4,12,4,4,14,4,6,4,8,8,4,4,18,4,4,8,8,4,12,4,22,4,4,22,4,
%T A362817 4,22,4,8,8,4,8,8,4,4,26,4,10,4,8,8,4,8,8,4,28,4,4,30,4,4,30
%N A362817 Irregular triangle read by rows: T(n,k) (n>=1, k>=1) is the number of edges of the k-th polygon (or part), from left to right, of the symmetric representation of sigma(n).
%C A362817 Row n is [4, 4] if and only if n is an odd prime.
%C A362817 If the symmetric representation of sigma(n) has only one polygon (or part), or in other words, if n is a member of A174973 (also of the same sequence A238443) then row n has only a term: T(n,1) = 2 + 2*(A003056(n-1) + A003056(n)). Note that A174973 = A238443 also include all powers of 2 and all even perfect numbers.
%e A362817 Triangle begins:
%e A362817    4;
%e A362817    6;
%e A362817    4,  4;
%e A362817   10;
%e A362817    4,  4;
%e A362817   12;
%e A362817    4,  4;
%e A362817   14;
%e A362817    4,  6,  4;
%e A362817    8,  8;
%e A362817    4,  4;
%e A362817   18;
%e A362817    4,  4;
%e A362817    8,  8;
%e A362817    4, 12,  4;
%e A362817   ...
%e A362817 Illustration of row 9:
%e A362817          4
%e A362817      _ _ _ _ _
%e A362817     |_ _ _ _ _|
%e A362817               |_ _ 6
%e A362817               |_  |
%e A362817                 |_|_ _
%e A362817                     | |
%e A362817                     | |
%e A362817                     | |  4
%e A362817                     | |
%e A362817                     |_|
%e A362817 .
%e A362817 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].
%Y A362817 Row lengths give A237271.
%Y A362817 Row sums give A362818.
%Y A362817 Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.
%K A362817 nonn,tabf,more
%O A362817 1,1
%A A362817 _Omar E. Pol_, May 04 2023