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All odd primes are in the sequence because the parts of the symmetric representation of sigma(prime(i)) are [m, m], where m = (1 + prime(i))/2, for i >= 2.

There are no odd composite numbers in this sequence.

First differs from A173708 at a(13).

Since sigma(p*q) >= 1 + p + q + p*q for odd p and q, the symmetric representation of sigma(p*q) has more parts than the two extremal ones of size (p*q + 1)/2; therefore, the above comments are true. - Hartmut F. W. Hoft, Jul 16 2014

From Hartmut F. W. Hoft, Sep 16 2015: (Start)

The following two statements are equivalent:

(1) The symmetric representation of sigma(n) has two parts, and

(2) n = q * p where q is in A174973, p is prime, and 2 * q < p.

For a proof see the link and also the link in A071561.

This characterization allows for much faster computation of numbers in the sequence - function a239929F[] in the Mathematica section - than computations based on Dyck paths. The function a239929Stalk[] gives rise to the associated irregular triangle whose columns are indexed by A174973 and whose rows are indexed by A065091, the odd primes. (End)

From Hartmut F. W. Hoft, Dec 06 2016: (Start)

For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime:

(a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p].

(b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p].

(c) each number k is of the form 2^m * p with p prime [by definition].

m = 1: (a) A100484 even semiprimes (except 4 and 6)

(b) A161344 (except 4, 6 and 8)

(c) A001747 (except 2, 4 and 6)

m = 2: (a) A270298

(b) A161424 (except 16, 20, 24, 28 and 32)

(c) A001749 (except 8, 12, 20 and 28)

m = 3: (a) A270301

(b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)

(c) sequence not in OEIS

b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End)

Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).

This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.

A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.

Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.

For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.

Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.