A360022 Triangle read by rows: T(n,k) is the sum of the widths of the k-th diagonals of the symmetric representation of sigma(n).

1, 1, 2, 0, 2, 2, 1, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Omar E. Pol, Jan 22 2023

Keywords

• nonn
• tabl

Comments

The main diagonal of the diagram called "symmetric representation of sigma(n)" is its axis of symmetry. In this case it is also the first diagonal of the diagram. The second diagonals are the two diagonals that are adjacent to the main diagonal. The third diagonals are the two diagonals that are adjacent to the second diagonals. And so on.

If and only if n is a power of 2 (A000079) then row n lists the first n terms of A040000 (the same sequence as the right border of the triangle).

If and only if n is an odd prime (A065091) then row n lists (n - 1)/2 zeros together with 1 + (n - 1)/2 2's.

If and only if n is an even perfect number (Cf. A000396) then row n lists n 2's (the first n terms of A007395).

For further information about the mentioned "widths" see A249351.

Examples

			Triangle begins (rows: 1..16):
  1;
  1, 2;
  0, 2, 2;
  1, 2, 2, 2;
  0, 0, 2, 2, 2;
  2, 2, 2, 2, 2, 2;
  0, 0, 0, 2, 2, 2, 2;
  1, 2, 2, 2, 2, 2, 2, 2;
  1, 2, 0, 0, 2, 2, 2, 2, 2;
  0, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2;
  2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2;
  0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2;
  1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  ...
		

Links

• Omar E. Pol, Illustration of initial terms of the column 1 = A067742
• Index entries for sequences related to sigma(n)

Crossrefs

Row sums give A000203.

Column 1 gives A067742.

Right border gives A040000.

Cf. A000079, A000396, A007395, A065091, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A262626.

Formula

T(n,1) = A067742(n) = A249351(n,n).

T(n,k) = 2*A249351(n,n+k-1), if 1 < k <= n.