A244363 -id:A244363 - cp's OEIS frontend

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).

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, 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

Offset: 1

Omar E. Pol, May 04 2023

Row n is [4, 4] if and only if n is an odd prime.

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.

			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].

Row lengths give A237271.
Row sums give A362818.
Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.

A362818 Total number of edges of all polygons (or parts) of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 8, 10, 8, 12, 8, 14, 14, 16, 8, 18, 8, 16, 20, 22, 8, 22, 8, 22, 24, 16, 8, 26, 18, 16, 24, 28, 8, 30, 8, 30

Offset: 1

Omar E. Pol, May 04 2023

a(n) = 8 if and only if n is an odd prime.

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 a(n) = 2 + 2*(A003056(n-1) + A003056(n)). Note that A174973 = A238443 also include all powers of 2 and all even perfect numbers.

			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.

Row sums of A362817.

Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.

cp's OEIS Frontend

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).

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