A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.

1, 4, 1, 8, 4, 1, 1, 15, 8, 4, 4, 1, 1, 1, 21, 15, 8, 8, 4, 4, 4, 1, 1, 1, 1, 1, 33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 41, 33, 21, 21, 15, 15, 15, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 41, 33, 33, 21, 21, 21, 15, 15, 15, 15, 15

Offset: 1

Author

Omar E. Pol, Jan 10 2021

Keywords

• nonn
• tabf

Comments

Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593).

The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).

T(n,k) is the volume (the number of cells) in the k-th level starting from the base.

This polycube has the property that the volume (the total number of cells) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.

A dissection of the symmetric tower is a three-dimensional spiral whose top view is described in A239660.

Other triangles related to the volume of this polycube are A340527 and A340579.

The symmetric tower is a member of the family of the stepped pyramid described in A245092.

For another symmetric tower of the same family and whose volume equals A066186(n) see A340423.

The sum of row n of triangle equals A182738(n). That property is due to the correspondence between divisors and parts. For more information see A336811.

Examples

			Triangle begins:
   1;
   4,  1;
   8,  4,  1,  1;
  15,  8,  4,  4, 1, 1, 1;
  21, 15,  8,  8, 4, 4, 4, 1, 1, 1, 1, 1;
  33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 5 the length of row 5 is A000070(4) = 12.
The sum of row 5 is 21 + 15 + 8 + 8 + 4 + 4 + 4 + 1 + 1 + 1 + 1 + 1 = 69, equaling A182738(5).
		

Crossrefs

Row sums give A182738.

Cf. A340527 (a regular version).

Members of the same family are: A176206, A337209, A339258, A340530.

Cf. A221529, A239660, A339278, A339304, A340423, A340529.

Cf. A000070, A024916, A237593, A336811, A338156.

Formula

a(m) = A024916(A176206(m)), assuming A176206 has offset 1.

T(n,k) = A024916(A176206(n,k)), assuming A176206 has offset 1.