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

1,3

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.

Links

Table of n, a(n) for n=1..105.

Omar E. Pol, Illustration of initial terms of the column 1 = A067742

Index entries for sequences related to sigma(n)

Formula

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

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

Example

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;
  ...

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.

Sequence in context: A023604 A219660 A060964 * A118206 A029314 A071635

Adjacent sequences: A360019 A360020 A360021 * A360023 A360024 A360025

Keyword

nonn,tabl

Author

Omar E. Pol, Jan 22 2023

Status

approved