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This sequence is a companion to A279387 in which each contiguous sequence of identical widths w in A249223 are replaced by a single entry of w. Using the resulting distribution pattern of widths across all parts of the symmetric representation of sigma(n) the subparts at each level are counted in A279387.

The sequence of widths are computed first to the diagonal of the symmetric representation of sigma only for those numbers in set F defined in A341971. Then the reversed list less its first number is appended so that the width at the diagonal is not listed twice. Thus every row contains an odd number of entries and is symmetric about its center entry.

Let 1 <= n, 1 <= d <= s = A001227(n) and 1 <= k <= r = floor((sqrt(8*n + 1) - 1)/2). Let Q(n,d) be row n in the triangle of A341970, R(n,d) be row n in the triangle of A341970 and S(n,d) = R(n,Q(n,d)), then T(n,e) = S(n,e) for 1 <= e <= s and T(n,e) = S(n,2*s - e) for s < e <= 2*s - 1 is row n for this sequence.

Conjecture 1: where records occur in A237271. - Omar E. Pol, Dec 27 2016

For more information about the symmetric representation of sigma see A237270, A237593.

This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - Hartmut F. W. Hoft, Dec 09 2014

Conjecture 2: all terms are odd numbers. - Omar E. Pol, Oct 14 2018

Proof of Conjecture 2: Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - Hartmut F. W. Hoft, Sep 22 2021

Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021

From Omar E. Pol, Jul 28 2025: (Start)

Conjecture 3: a(n) is the smallest number k having n 2-dense sublists of divisors of k.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.

An example of the conjecture 3 for n = 1..5 is as shown below:

----------------------------------------------------

| | List of divisors of k | | |

| k | [with sublists in brackets] | n | a(n) |

----------------------------------------------------

| 1 | [1]; | 1 | 1 |

| 3 | [1], [3]; | 2 | 3 |

| 9 | [1], [3], [9]; | 3 | 9 |

| 21 | [1], [3], [7], [21]; | 4 | 21 |

| 63 | [1], [3], [7, 9], [21], [63]; | 5 | 63 |

(End)

Conjecture 4: a(n) is the smallest number k having n divisors p of k such that p is greater than twice the adjacent previous divisor of k. - Omar E. Pol, Aug 05 2025

This is a permutation of the positive integers.

All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.

The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018

Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022

Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

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

The sequence is not increasing with the maximum width of the symmetric representation just like A347979.

Observation: a(2)..a(14) ending in 5. - Omar E. Pol, Sep 23 2021

For the 30 known terms the symmetric representation of sigma consists of a single part, i.e., this is a subsequence of A174973 = A238443.

The sequence is not increasing with the maximum width of the symmetric representation of sigma.

Also a(33) = 2162160 is the only further number in the sequence less than 2500000.

The first row of the table below is A318843 and the first column is A250070.

T(1,k+1) <= 3^k, for all k>=0, since for k=2j the (j+1)-st part in the symmetric representation of sigma(3^k) extends across the diagonal, and for k=2j+1 the (j+1)-st part is completed before the diagonal.

The data computed so far for a partially filled table of 15 rows and 15 columns, show that all rows, all columns (except column 4 for n <= 6 *10^7), and the diagonal are nonmonotonic.

Theorem of correspondence between the partitions of n into k consecutive parts and the odd divisors of n: given a partition of n into k consecutive parts if k is odd then the corresponding odd divisor of n is k, otherwise if k is even then the corresponding odd divisor of n is the sum of any pair of conjugate parts of the partition (for example the sum of the largest part and the smallest part).

Conjecture: the first A001227(n) terms in the n-th row are also the absolute values of the n-th row of A341971.

The last A001227(n) terms in the n-th row are also the mirror of the n-th row of A261697.