cp's OEIS Frontend

The diagram of the symmetry of sigma has been via A196020 --> A236104 --> A235791 --> A237591 --> A237593.

For more information see A237270.

a(n) is also the number of terraces at n-th level (starting from the top) of the stepped pyramid described in A245092. - Omar E. Pol, Apr 20 2016

a(n) is also the number of subparts in the first layer of the symmetric representation of sigma(n). For the definion of "subpart" see A279387. - Omar E. Pol, Dec 08 2016

Note that the number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n. (See the second example). - Omar E. Pol, Dec 20 2016

From Hartmut F. W. Hoft, Dec 26 2016: (Start)

Using odd prime number 3, observe that the 1's in the 3^k-th row of the irregular triangle of A237048 are at index positions

3^0 < 2*3^0 < 3^1 < 2*3^1 < ... < 2*3^((k-1)/2) < 3^(k/2) < ...

the last being 2*3^((k-1)/2) when k is odd and 3^(k/2) when k is even. Since odd and even index positions alternate, each pair (3^i, 2*3^i) specifies one part in the symmetric representation with a center part present when k is even. A straightforward count establishes that the symmetric representation of 3^k, k>=0, has k+1 parts. Since this argument is valid for any odd prime, every positive integer occurs infinitely many times in the sequence. (End)

a(n) = number of runs of consecutive nonzero terms in row n of A262045. - N. J. A. Sloane, Jan 18 2021

Indices of odd terms give A071562. Indices of even terms give A071561. - Omar E. Pol, Feb 01 2021

a(n) is also the number of prisms in the three-dimensional version of the symmetric representation of k*sigma(n) where k is the height of the prisms, with k >= 1. - Omar E. Pol, Jul 01 2021

With a(1) = 0; a(n) is also the number of parts in the symmetric representation of A001065(n), the sum of aliquot parts of n. - Omar E. Pol, Aug 04 2021

The parity of this sequence is also the characteristic function of numbers that have middle divisors. - Omar E. Pol, Sep 30 2021

a(n) is also the number of polycubes in the 3D-version of the ziggurat of order n described in A347186. - Omar E. Pol, Jun 11 2024

Conjecture 1: a(n) is the number of odd divisors of n except the "e" odd divisors described in A005279. Thus a(n) is the length of the n-th row of A379288. - Omar E. Pol, Dec 21 2024

The conjecture 1 was checked up n = 10000 by Amiram Eldar. - Omar E. Pol, Dec 22 2024

The conjecture 1 is true. For a proof see A379288. - Hartmut F. W. Hoft, Jan 21 2025

From Omar E. Pol, Jul 31 2025: (Start)

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

We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.

In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.

Example: for n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10], so a(10) = 2.

The conjecture 2 is essentially the same as the second conjecture in the Comments of A384149. See also Peter Munn's formula in A237270.

The indices where a(n) = 1 give A174973 (2-dense numbers). See the proof there. (End)

Conjecture 3: a(n) is the number of divisors p of n such that p is greater than twice the adjacent previous divisor of n. The divisors p give the n-th row of A379288. - Omar E. Pol, Aug 02 2025

The "subparts" of the symmetric representation of sigma(n) are defined to be the regions that arise after the dissection of the symmetric representation of sigma(n) into successive layers of width 1.

The number of layers of width 1 in the symmetric representation of sigma(n) is given in A250068.

The number of subparts in the first layer of the symmetric representation of sigma(n) is equal to A237271(n).

We can find the symmetric representation of sigma(n) as the terraces at the n-th level (starting from the top) of the stepped pyramid described in A245092.

(All above comments are essentially the same as the comments dated Nov 05 2016 at the old version of A275601, which was the same as A001227).

The sum of row n equals the number of subparts in the symmetric representation of sigma(n).

Conjecture:

The number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n.

From Hartmut F. W. Hoft, Dec 16 2016: (Start)

Proof:

Each row of the irregular triangle of A262045 can be interpreted as a step function of step sizes 1, 0, and -1. The numbers in row n are the widths of the segments in the parts of the symmetric representation of sigma(n). Each new subpart in a segment (in the left half) of row n starts at the same odd index that represents an odd divisor d of n in the irregular triangle of A237048. Either a subpart ends at an even index e, representing a second odd divisor, which satisfies d * e = oddpart(n), and thus the entire subpart is duplicated in the symmetric portion of the representation, or a subpart runs through the center and continues contiguously into the right half of the symmetric portion of the representation. In other words, the number of subparts in row n equals the number of odd divisors of n, i.e., the conjecture is true. (End)

Conjecture: row n is formed by the odd-indexed terms of the n-th row of triangle A280850 together with the even-indexed terms of the same row but listed in reverse order. Examples: the 15th row of A280850 is [8, 8, 7, 0, 1] so the 15th row of this triangle is [8, 7, 1, 0, 8]. The 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0] so the 75h row of this triangle is [38, 21, 3, 0, 0, 0, 21, 0, 3, 0, 38].

For the definition of "subparts" see A279387.

For more information about the mentioned Dyck paths see A237593.

T(n,k) could be called the "charge" of the k-th peak of the largest Dyck path of the symmetric representation of sigma(n).

The number of zeros in row n is A238005(n). - Omar E. Pol, Sep 11 2021

For the construction of the n-th row of this triangle start with a copy of the n-th row of the triangle A196020.

Then replace each element of the m-th pair of positive integers (x, y) with the value (x - y)/2, where "y" is the m-th even-indexed term of the row, and "x" is its previous nearest odd-indexed term not used in another pair in the same row, if such a pair exist. Otherwise T(n,k) = A196020(n,k). (See example).

Observation 1: at least for the first 28 rows of the triangle the nonzero terms in the n-th row are also the subparts of the symmetric representation of sigma(n), assuming the ordering of the subparts in the same row does not matter.

Question 1: are always the nonzero terms of the n-th row the same as all the subparts of the symmetric representation of sigma(n)? If not, what is the index of the row in which appears the first counterexample?

Note that the "subparts" are the regions that arise after the dissection of the symmetric representation of sigma(n) into successive layers of width 1.

For more information about "subparts" see A279387 and A237593.

About the question 1, it appears that the n-th row of the triangle A280851 and the n-th row of this triangle contain the same nonzero numbers, though in different order; checked through n = 250000. - Hartmut F. W. Hoft, Jan 31 2018

From Omar E. Pol, Feb 02 2018: (Start)

Observation 2: at least for the first 28 rows of the triangle we have that in the n-th row the odd-indexed terms, from left to right, together with the even-indexed terms, from right to left, form a finite sequence in which the nonzero terms are the same as the n-th row of triangle A280851, which lists the subparts of the symmetric representation of sigma(n).

Question 2: Are always the same for all rows? If not, what is the index of the row in which appears the first counterexample? (End)

Conjecture: the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A296508 (this is the same conjecture from A296508). - Omar E. Pol, Apr 20 2018

The terms in the n-th row are the same as the terms in the n-th row of triangle A279391, but in some rows the terms appear in distinct order.

First differs from A279391 at a(28) = T(15,3).

Also nonzero terms of A296508. - Omar E. Pol, Feb 11 2018

Since the width of the single region of the symmetric representation of sigma( 2^ceiling((p-1)*(log_2 3) - 1) * 3^(p-1) ), for prime number p, at the diagonal equals p, this sequence contains an increasing subsequence (see A250071).

a(n) is also the number of layers of width 1 in the symmetric representation of sigma(n). For more information see A001227. - Omar E. Pol, Dec 13 2016

The original name was: Number of odd divisors of n minus the number of parts in the symmetric representation of sigma(n).

Observation: at least the indices of the first 42 positive elements coincide with A005279: 6, 12, 15, 18, 20, 24..., checked (by hand) up to n = 2^7.

The observation is true for the indices of all positive elements. Hence the indices of the zeros give A174905. - Omar E. Pol, Jan 06 2017

a(n) is the number of subparts minus the number of parts in the symmetric representation of sigma(n). For the definition of "subpart" see A279387. - Omar E. Pol, Sep 26 2018

a(n) is the number of subparts of the symmetric representation of sigma(n) that are not in the first layer. - Omar E. Pol, Jan 26 2025

Conjecture 1: the number of nonzero terms in row n equals A082647(n).

Conjecture 2: column k lists positive integers interleaved with 2*k+2 zeros.

The subparts of the ziggurat diagram are the polygons formed by the cells that are under the staircases.

The connection of the subparts of the ziggurat diagram with the polygonal numbers is as follows:

The area under a double-staircase labeled with the number j is equal to the m-th (j+2)-gonal number plus the (m-1)-th (j+2)-gonal number, where m is the number of steps on one side of the ladder from the base to the top.

The area under a simple-staircase labeled with the number j is equal to the m-th (j+2)-gonal number, where m is the number of steps.

So the k-th column of the triangle is related to the (2*k+1)-gonal numbers, for example:

For the calculation of column 1 we use triangular numbers A000217.

For the calculation of column 2 we use pentagonal numbers A000326.

For the calculation of column 3 we use heptagonal numbers A000566.

For the calculation of column 4 we use enneagonal numbers A001106.

And so on.

More generally, for the calculation of column k we use the (2*k+1)-gonal numbers.

For further information about the ziggurat diagram see A347186.

The "subparts" of the symmetric representation of sigma(n) are the regions that arise after the dissection of the symmetric representation of sigma(n) into successive layers of width 1.

The number of subparts in the symmetric representation of sigma(n) equals the number of odd divisors of n.

For more information about "subparts" see A279387, A279388 and A279391.

Note that we can find the symmetric representation of sigma(n) as the terraces at the n-th level (starting from the top) of the stepped pyramid described in A245092.