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Here T(n,k) is defined to be the "k-th width" of the symmetric representation of A024916(n), with n>=1 and 1<=k<=2n-1.

If both A249351 and this sequence are written as isosceles triangles then the partial sums of the columns of A249351 give the columns of this isosceles triangle (see the second triangle in Example section).

For the definition of the k-th width of the symmetric representation of sigma(n) see A249351.

Note that for the geometric representation of the n-th row of the triangle we need the x-axis, the y-axis, and only a Dyck path which is given by the elements of the n-th row of the triangle A237593.

Row n has length 2*n-1.

Row sums give A024916.

The middle diagonal is A240542.

The sequence is the "diagonal" - first element in each column - of the triangle of numbers associated with the symmetric representation of sigma(n) when it has two parts, each of width one (see A246955).

The areas of the first two regions are (2^(m+1) - 1) * (p * q + 1) / 2 and (2^(m+1) - 1) * (p + q) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (q + 1).

For a proof of the formula for this sequence see the link.

This sequence is a permutation of A174905. Numbers in the even numbered columns of the table form A241008 and those in the odd numbered columns form A241010. The first row of the table is A318843.

This sequence is a subsequence of A240062 and each column in this sequence is a subsequence in the respective column of A240062.

Also indices of the powers of 2 in A033676.

Also numbers in increasing order from the columns k of A163280 where k is a power of 2.

Observation: at least the first 82 terms of the subsequence of terms with no middle divisors (that is 3, 5, 7, 10, ...) coincide with at least the first 82 terms of A246955.

For the definition of middle divisor see A067742.

From Peter Munn, Oct 26 2023: (Start)

Most of the early terms are in A342081, which consists of powers of 2 together with products of a prime and a power of 2 where the prime is the larger. The exceptions are 24, 72, 80, 96, 112, ... .

The odd terms clearly consist of 1 and the odd primes. We can fully characterize the even terms by their A290110 values, which depend on the relative sizes of a number's divisors. A290110 provides a refinement of the classification of numbers by prime signature (cf. A212171): see the example below for numbers with the same prime signature as 48.

(End)

For the definition of k-th width of the symmetric representation of sigma(n) see A249351.

Row n list the first n terms of the n-th row of A249351.

It appears that the leading diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).

For more information see A237591, A237593.

Note that odd primes (A065091) are also the numbers j with the property that the parts of the symmetric representation of sigma(j) are two rectangles or more generally two quadrilaterals.

Conjecture 1: The octagons are S-shaped and they have width 1.

Conjecture 2: This sequence is also the primes doubled (or even semiprimes) >= 10 (Cf. A100484). - Omar E. Pol, Aug 15 2023

For the symmetric representation of sigma(n) to consist of 2 octagons the first 3 entries in row n of the triangle of A249223 must be nonzero, hence must be 1's, indicating width 1, with the remaining entries zero. Therefore, row n of A237048 is 100100..., implying n = 2*p with p>3 prime. Both conjectures are true. - Hartmut F. W. Hoft, Aug 22 2023

From Omar E. Pol, Aug 23 2023: (Start)

Also the row numbers of the triangle A364639 where the rows are [1, 0, -1, 1] or where the rows start with [1, 0, -1, 1] and the remaining terms are zeros.

Each supersequence A063221 >= 10 and A091999 >= 10 gives the numbers k with the property that the first part of the symmetric representation of sigma(k) is an octagon. In that case each supersequence gives the row numbers of the triangle A364639 where the rows start with [1, 0, -1]. (End)

This sequence is a subsequence of A264102 and also of A014105, the second hexagonal numbers. Every number in this sequence is a triangular number.

The sequence A156592 of products of a Sophie Germain prime (A005384) and its associated safe prime (A005385) except for the first pair (2, 5) forms a subsequence of this sequence, the first column in the irregular triangular grid in the example.

The areas of the first two regions are (2^(m+1) - 1) * (2^(m+1) * p^2 * p + 1) / 2 and (2^(m+1) - 1) * (2^(m+1) * p + p + 1) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (2^(m+1) * p + 2).

For a proof of the formula for this sequence see the link.

First differs from A071562 at a(12) = 21 here, there a(12) = 24.

For column k = 1, 2, 3, 4, 5, ... the number of sides of the mentioned s-gon are respectively 4, 8, 16, 32, 64, ...

Conjecture 1: column k gives the row numbers of the triangle A364639 where the rows are [1, A036563(k+1) zeros, -1, 1] or where the rows start with [1, A036563(k+1) zeros, -1, 1] and the remaining terms are zeros.

Conjecture 2: every column gives a subsequence of A246955.

Conjecture 3: the sequence is infinite.

Observation 1: at least the terms <= 199 in increasing order coincide with at least the first 82 terms of the intersection of A071561 and A365406.

Observation 2: in the Example section of A246955 there is an irregular triangle. It seems that the terms sorted of the triangle give the sequence A246955. At least the first r(k) terms in the column (k - 1) of the triangle coincide with the first r(k) terms of the column k of this square array, where r(k) are 19, 18, 16, 14, 7 for k = 1..5 respectively.

Observation 3: at least the first five terms of the row 1 coincide with the first five terms of A246956.