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Partial sums of A244971.

If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.

If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).

a(n) is also the total number of terraces of the stepped pyramid with n levels described in A244050. - Omar E. Pol, Apr 20 2016

Since the diagram is symmetric so all terms are even numbers.

For another version with subparts see A340848 from which first differs at a(6).

Row n lists the parts of the symmetric representation of A008438(n-1).

Also these are the parts from the odd-indexed rows of A237270.

Also these are the parts in the quadrants 1 and 3 of the spiral described in A239660, see example.

Row sums give A008438.

The length of row n is A237271(2n-1).

Both column 1 and the right border are equal to n.

Note that also the sequence can be represented in a quadrant.

We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

Conjecture: a(n) is the number of divisors r of n such that r is not greater than twice the adjacent previous divisor of n. - Omar E. Pol, Aug 04 2025

Numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.

Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

In order to construct this sequence and the diagram of the symmetric representation of antisigma(n) = A024816(n) we use the following rules:

At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593. The area of the region that is below the symmetric representation of sigma(n) equals A024916(n-1).

At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n). Then we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals A244048(n) = A153485(n-1). The area of the region that is below the staircase equals A000217(n-1).

At stage 3 we turn OFF the cells of the symmetric representation of sigma(n) and also the cells that are below the staircase. Then we turn ON the rest of the cells that are in the square n X n. The result is that the ON cell form the diagram of the symmetric representation of antisigma(n) = A024816(n). See the Example section.

For n >= 7; if A237271(n) = 1 or n is a term of A262259 then a(n) = 2 otherwise a(n) = 1.

For more information see A239663 and A239665.

For the definition of "part" of the symmetric representation of sigma see A237270.

For more information about the mentioned Dyck paths see A237593.

Sum of divisors of the numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.

Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

Conjecture 1: there are infinitely many pairs of the form a(x) = y; a(y) = x (see examples).

First differs from A351904 at a(11).

From Hartmut F. W. Hoft, Jun 10 2024: (Start)

For numbers less than or equal to a(2^20), (2^k, 2^(k+1) - 1), 0 <= k <= 19, are the only pairs satisfying a(a(x)) = x; the triple (36, 46, 91) is the only one satisfying a(a(a(x))) = x, and there are no proper order 4 quadruples and no order 5 quintuples, apart from fixed point 1.

Conjecture 2: Only the pairs x = 2^k and y = 2^(k+1) - 1, k >= 0, satisfy a(x) = y and a(y) = x.

A repeated number d in this sequence determines a pair of distinct indices u and v such that d = a(u) = a(v). This means that d is the smallest number for which parts of sizes u and v occur in the symmetric representation of sigma(d), SRS(d). There are 5507 such pairs less than a(2^20). (End)