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Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the area of the diagram (see the example). For more information see A237593 and its related sequences.

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

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.

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)

This property of a(n) is because the symmetric representation of sigma(a(n)+1) has only one part.

All terms are odd.

First differs from A085493 at a(22).

Also primes p such that both Dyck paths of the symmetric representation of sigma(p) have a central peak.

Note that the symmetric representation of sigma of an odd prime consists of two perpendicular bars connected by an irregular zig-zag path (see example).

Odd primes and the terms of this sequence are easily identifiable in the pyramid described in A245092 (see Links section).

For more information about the mentioned Dyck paths see A237593.

Equivalently, primes p such that the largest Dyck path of the symmetric representation of sigma(p) has an odd number of peaks.

Except for the first term 3, primes p such that both Dyck paths of the symmetric representation of sigma(p) have a central valley.

Note that the symmetric representation of sigma of an odd prime consists of two perpendicular bars connected by an irregular zig-zag path (see example).

Odd primes and the terms of this sequence are easily identifiable in the pyramid described in A245092 (see Links section).

For more information about the mentioned Dyck paths see A237593.

Equivalently, primes p such that the largest Dyck path of the symmetric representation of sigma(p) has an even number of peaks.

a(n) is the volume (or the number of cubes) in a polycube whose base is the symmetric representation of A024916(n) which is formed with the first n 3D-Ziggurats described in A347186.

a(n) is also the total number of cubes in a three-dimensional spiral formed with the first n 3D-Ziggurats described in A347186 (see example). The base of the 3D-spiral is the spiral formed with the symmetric representation of sigma of the first n positive integers as shown in the example section of A239660.