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Let n = 2^m * q with m >= 0 and q odd, let row(n) = floor(sqrt(8*n+1) - 1)/2), and let 1 = d_1 < ... < d_h <= row(n) < d_(h+1) < ... < d_k = q be the k odd divisors of n.

The symmetric representation of sigma(n) consists of 3 parts precisely when there is a unique i, 1 <= i < h, such that 2^(m+1) * d_i < d_(i+1) and d_h <= row(n) < 2^(m+1) * d_h.

This property of the odd divisors of n is equivalent to the n-th row of the irregular triangle of A249223 consisting of a block of positive numbers, followed by a block of zeros, followed in turn by a block of positive numbers, i.e., determining the first part and the left half of the center part of the symmetric representation of sigma(n), resulting in 3 parts.

Let n be the product of two primes p and q satisfying 2 < p < q < 2*p. Then n satisfies the property above so that the odd numbers in A087718 form a subsequence.

The sequence is infinite since, for example, for any n >= 1 the symmetric representation of sigma(3^n) consists of n + 1 parts of width 1. However, it is not increasing since a(11) = 59049 = 3^10 and a(12) = 29095 = 5 * 11 * 23^2. Also a(13) <= 531441 = 3^12.

This sequence is a subsequence of A174905; its subsequences a(n) for odd/even n are subsequences of A241010/A241008, respectively. Some even-indexed elements of this sequence are members of A239663, e.g., a(2), a(4), a(6), a(8) and a(12), but not a(10) = 6875.

The central pair of parts in the symmetric representation of sigma(a(2)), sigma(a(4)) and sigma(a(8)) meets at the diagonal (see A298856).

From Hartmut F. W. Hoft, Oct 04 2021: (Start)

An upper bound to the sequence is a(n) <= 3^(n-1), n >= 1, (see A348171).

For p = 1,2,3,5,7,11,13,17, a(p) = 3^(p-1) and this equality possibly holds for all a(p) with p a prime.

Also, 75 * 10^6 < a(19) <= 3^18, a(20) = 15391255, a(21) = 44289025 and a(n) > 75 * 10^6 for n > 21.

a(13)-a(18) computations based on A348171 rather than A237270.

The symmetric representation of sigma(3^(p-1)), p prime, consists of p parts and its middle part has area 3^((p-1)/2). (End)

a(n) >= A038547(n) with equality for n=1 and primes n since the distinct prime divisors of n can be replaced by primes 3, 5, 7, 11, ... yielding a smaller number k with the same number of odd divisors. However, some parts in the symmetric representation of sigma(k) have width at least 2. - Hartmut F. W. Hoft, Dec 11 2023

Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 5 in the table of A357581. - Hartmut F. W. Hoft, Oct 04 2022

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.

Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 6 in the table of A357581. - Hartmut F. W. Hoft, Oct 04 2022

Numbers n such that A243982(n) = 0.

First differs from A151991 at a(25).

Let n = 2^m * q with m >= 0 and q odd. Let c_n denote the count of regions in the symmetric representation of sigma(n), which is determined by the positions of 1's in the n-th row of A237048. The maximum of c_n occurs when odd and even positions of 1's alternate implying that all regions have width 1, denoted by w_n = 1. When m > 0 then sigma_0(n) > sigma_0(q) and c_n = sigma_0(n) is impossible. Therefore, exactly those odd n with w_n = 1 are in this sequence. Furthermore, since the 1's in A237048 represent the odd divisors of n, their odd-even alternation expresses the property 2*f < g for any two adjacent divisors f < g of odd number n; in other words, this sequence is also the complement of A090196 relative to the odd numbers. This last property permits computations of elements in this sequence faster than with function a244579, which is based on Dyck paths. - Hartmut F. W. Hoft, Oct 11 2015

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

Also, integers n such that for any pair a < b of divisors of n the inequality 2*a < b holds (hence n is odd).

Let 1 = d_1 < ... < d_k = n be all (odd) divisors of n. The property 2*d_i < d_(i+1), for 1 <= i < k, is equivalent for the 1's in the n-th row of A249223 to be in positions 1 = d_1 < 2 < d_2 < 2*d_2 < ... < d_i <2*d_i < d_(i+1) < ... where 2*d_i represents the odd divisor e_i with d_i * e_i = n. In other words, the odd divisors are the number of parts in the symmetric representation of sigma(n). The rightmost 1 in the n-th row occurs in an odd (even) position when k is odd (even).

As a consequence this sequence is also the complement of A090196 in the set of odd numbers. (End)

For more information see A239663 and A239665.

It appears that a(n) = 2 * q where q is odd and that the symmetric representation of sigma(a(n)/2) has the same number of parts as that for a(n). Number a(12) > 15000000. - Hartmut F. W. Hoft, Sep 22 2021

This is a permutation of the positive even numbers (A299174).

The union of all odd-indexed columns gives A319796, the even numbers in A071562.

The union of all even-indexed columns gives A319802, the even numbers in A071561.

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

Observation: the first 29 terms coincide with the first 29 terms of A161345 that are >= 21.

Apparently a(n)=A127329(n) for n>2. - R. J. Mathar, Sep 05 2023