A174973 -id:A174973 - cp's OEIS frontend

A323648 Numbers k such that the largest Dyck path of the symmetric representation of sigma(k) does not share any line segment with the largest Dyck path of the symmetric representation of sigma(k+1).

1, 2, 3, 5, 7, 9, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 47, 53, 55, 59, 63, 65, 71, 77, 79, 83, 87, 89, 95, 99, 103, 107, 111, 119, 125, 127, 131, 135, 139, 143, 149, 155, 159, 161, 167, 175, 179, 191, 195, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239, 251, 255

Offset: 1

Omar E. Pol, Apr 02 2019

nonn

Equivalently, numbers k such that in the perspective view of the stepped pyramid described in A245092, the steps of the n-th level do not share any vertical face with the steps of the level n + 1, starting from the top of the pyramid.
a(2) = 2 is the only even number in the sequence.
For more information about the Dyck paths, the connection with the sum of divisors function A000203, and the connection with the theory of partitions see A237593.

Cf. A174973, A000203, A196020, A236104, A237270, A237271, A237591, A237593, A244145, A245092, A245192, A262259, A262626, A279029, A279244.

(* Function path[] is defined in A237270 *)
a323648Q[n_] := Length[Select[Transpose[{Take[path[n+1], {2,-2}], path[n]}], #[[1]]==#[[2]]&]]<=1
a323648[n_] := Select[Range[n], a323648Q]
a323648[255]
(* Functions a262259Q[ ] and a174973Q[ ] are defined in A279029 *)
a323648[n_] := Select[Range[n], a262259Q[#+1]||a174973Q[#+1]&]
a323648[255] (* Hartmut F. W. Hoft, Jan 25 2025 *)

a(n) = A279029(n+1) - 1, for n >= 1. - Hartmut F. W. Hoft, Jan 25 2025

a(17)-a(63) by Hartmut F. W. Hoft, Jan 25 2025

A347980 a(n) is the smallest odd number k whose symmetric representation of sigma(k) has maximum width n.

Original entry on oeis.org

1, 15, 315, 2145, 3465, 17325, 45045, 51975, 225225, 405405, 315315, 765765, 1576575, 2297295

Offset: 1

Hartmut F. W. Hoft, Sep 22 2021

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

			The pattern of maximum widths of the parts in the symmetric representation of sigma for the first four terms in the sequence is:
   a(n) parts  successive widths
     1:   1          1
    15:   3        1 2 1
   315:   3        1 3 1
  2145:   7    1 2 3 4 3 2 1

Cf. A174973, A237048, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A250070, A262045, A341969, A341970, A341971, A347979.

a262045[n_] := Module[{a=Accumulate[Map[If[Mod[n - # (#+1)/2, #]==0, (-1)^(#+1), 0] &, Range[Floor[(Sqrt[8n+1]-1)/2]]]]}, Join[a, Reverse[a]]]
a347980[n_, mw_] := Module[{list=Table[0, mw], i, v}, For[i=1, i<=n, i+=2, v=Max[a262045[i]]; If [list[[v]]==0, list[[v]]=i]]; list]
a347980[2500000,14] (* long evaluation time *)

A348705 a(n) is the total length of all line segments in the symmetric representation of sigma(n).

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 24, 32, 34, 40, 36, 48, 42, 54, 56, 64, 54, 72, 60, 80, 78, 82, 72, 96, 84, 96, 98, 112, 90, 120, 96, 128

Offset: 1

Omar E. Pol, Oct 30 2021

a(n) is also the number of toothpicks of length 1 needed to represent the symmetric representation of sigma(n) (see the examples).
The diagram is symmetric thus all terms are even.
If the symmetric representation of sigma(n) has only one part (cf. A174973) or if it has two parts and they meet at the center of the Dyck path (cf. A262259) then a(n) = 4*n, otherwise a(n) < 4*n. In other words: if n is a term of A279029 then a(n) = 4*n, otherwise a(n) < 4*n.

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      8       12         16          18              24
.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          24                   32                     34
.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   8     |_ _| | |  _
                _ _|_| | |  _
   3  12       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  16         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5  18           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  24             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  24               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  32                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  34                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  40                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  36                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  48                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  42                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  54                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  56                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  64                                 |_ _ _ _ _ _ _ _ _|
...

Cf. A008586 (upper bounds).
Cf. A237271 (number of parts or regions).
Cf. A340833 (number of vertices).
Cf. A340846 (number of edges).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A139250, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A245092, A262259, A262626, A279029, A279228, A348854.

a(n) = 2*A348854(n).

a(n) = A008586(n) - A279228(n). - Omar E. Pol, Dec 13 2021

A387030 Irregular triangle read by rows: T(n,k) is the number of primes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 3, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Aug 13 2025

In a 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.
The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
It is conjectured that row lengths are given by A237271.

			Triangle begins:
  0;
  1;
  0, 1;
  1;
  0, 1;
  2;
  0, 1;
  1;
  0, 1, 0;
  1, 1;
  0, 1;
  2;
  0, 1;
  1, 1;
  0, 2, 0;
  ...
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] and [5, 10]. There is a prime number in each sublist, so row 10 is [1, 1].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. Only the second sublist contains primes, so row 15 is [0, 2, 0].

Paolo Xausa, Table of n, a(n) for n = 1..12242 (rows 1..4000 of triangle, flattened).

Row sums give A001221.

Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930.

A387030row[n_] := Map[Count[#, _?PrimeQ] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A387030row, 50] (* Paolo Xausa, Aug 19 2025 *)

A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.

Original entry on oeis.org

2, 1, 6, 2, 1, 1, 11, 4, 3, 1, 1, 1, 19, 6, 4, 2, 2, 1, 1, 1, 28, 10, 5, 3, 3, 2, 1, 1, 1, 1, 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1, 53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1, 69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 17 2021

The characteristic shape of the symmetric representation of sigma(A014105(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.
T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k + 1 consecutive parts.
1 together with the first column gives A317186. - Michel Marcus, Jan 12 2025

			Triangle begins:
   2,  1;
   6,  2,  1, 1;
  11,  4,  3, 1, 1, 1;
  19,  6,  4, 2, 2, 1, 1, 1;
  28, 10,  5, 3, 3, 2, 1, 1, 1, 1;
  40, 13,  7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
  53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1;
  69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
  86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column h gives the n-th second hexagonal number (A014105).
Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
--------------------------------------------------------------------------------------
  n   h   S   Diagram
--------------------------------------------------------------------------------------
                  _             _                     _                             _
                 | |           | |                   | |                           | |
              _ _|_|           | |                   | |                           | |
  1   3   4  |_ _|1            | |                   | |                           | |
               2               | |                   | |                           | |
                            _ _| |                   | |                           | |
                           |  _ _|                   | |                           | |
                        _ _|_|                       | |                           | |
                       |  _|1                        | |                           | |
              _ _ _ _ _| | 1                         | |                           | |
  2  10  18  |_ _ _ _ _ _|2                          | |                           | |
                   6                          _ _ _ _|_|                           | |
                                             | |                                   | |
                                            _| |                                   | |
                                           |  _|                                   | |
                                        _ _|_|                                     | |
                                    _ _|  _|1                                      | |
                                   |_ _ _|1 1                                      | |
                                   |  3                               _ _ _ _ _ _ _| |
                                   |4                                |    _ _ _ _ _ _|
              _ _ _ _ _ _ _ _ _ _ _|                                 |   |
  3  21  32  |_ _ _ _ _ _ _ _ _ _ _|                              _ _|   |
                       11                                        |       |
                                                                _|    _ _|
                                                               |     |
                                                            _ _|    _|
                                                        _ _|      _|
                                                       |        _|1
                                                  _ _ _|    _ _|1 1
                                                 |         | 2
                                                 |  _ _ _ _|2
                                                 | |   4
                                                 | |
                                                 | |6
                                                 | |
              _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4  36  91  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                               19
.

Row sums give A014105, n >= 1.
Row lengths give A005843.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
For the characteristic shape of sigma(A174973(n)) see A317305.
Cf. A000203, A237591, A237593, A245092, A249351, A262626, A317186.

row(n) = my(m=n*(2*n + 1)); vector((sqrtint(8*m+1)-1)\2, k, ceil((m+1)/k - (k+1)/2) - ceil((m+1)/(k+1) - (k+2)/2)); \\ Michel Marcus, Jan 12 2025

A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

Original entry on oeis.org

2, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440

Offset: 1

Hartmut F. W. Hoft, Sep 22 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 pattern of maximum widths within the single part of the symmetric representation of sigma for the first four numbers in the sequence is:
  a(n) parts successive widths
    2:   1           1
    6:   1         1 2 1
   60:   1     1 2 3 2 3 2 1
  120:   1     1 2 3 4 3 2 1

Cf. A174973, A237048, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A250070, A262045, A341969, A341970, A341971, A347980.

a262045[n_] := Module[{a=Accumulate[Map[If[Mod[n - # (#+1)/2, #]==0, (-1)^(#+1), 0] &, Range[Floor[(Sqrt[8n+1]-1)/2]]]]}, Join[a, Reverse[a]]]
a347979[n_, mw_] := Module[{list=Table[0, mw], i, v}, For[i=2, i<=n, i+=2, v=Max[a262045[i]]; If [list[[v]]==0, list[[v]]=i]]; list]
a347979[2500000, 33] (* computes a(1..30), a(33); a(31..32) > 2500000 *)

It appears that a(n) = A250070(n) if n >= 2.

A363122 Numbers k such that the highest power of 2 dividing k is larger than the highest power of p dividing k for any odd prime p.

Original entry on oeis.org

2, 4, 8, 12, 16, 24, 32, 40, 48, 56, 64, 80, 96, 112, 120, 128, 144, 160, 168, 176, 192, 208, 224, 240, 256, 280, 288, 320, 336, 352, 384, 416, 448, 480, 512, 528, 544, 560, 576, 608, 624, 640, 672, 704, 720, 736, 768, 800, 832, 840, 864, 880, 896, 928, 960, 992

Offset: 1

Amiram Eldar, May 16 2023

Numbers k such that A006519(k) = A034699(k).

If k is a term of this sequence then k*2^m is a term for any m >= 0. The primitive terms are in A363123.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006519, A034699, A116882, A363123.

Subsequence of A174973.

q[n_] := Module[{e = IntegerExponent[n, 2]}, 2^e > Max[Power @@@ FactorInteger[n/2^e]]]; Select[Range[1000], q]

is(n) = {my(e = valuation(n, 2), m = n>>e); if(m == 1, n>1, f = factor(m); 2^e > vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); }

from itertools import count, islice
from sympy import factorint
def A363122_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:n&-n>max((p**e for p, e in factorint(n>>(~n&n-1).bit_length()).items()),default=0),count(max(startvalue,2)))
A363122_list = list(islice(A363122_gen(),20)) # Chai Wah Wu, May 17 2023

A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 3, 8, 1, 5, 36, 1, 7, 64, 1, 3, 9, 2, 50, 1, 11, 1728, 1, 13, 2, 98, 1, 15, 15, 1024, 1, 17, 5832, 1, 19, 8000, 1, 3, 7, 21, 2, 242, 1, 23, 331776, 1, 5, 25, 2, 338, 1, 3, 9, 27, 21952, 1, 29, 810000, 1, 31, 32768, 1, 3, 11, 33, 2, 578, 1, 35, 35, 10077696, 1, 37, 2, 722, 1, 3, 13, 39, 2560000

Offset: 1

Omar E. Pol, Aug 12 2025

In a 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.
The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
It is conjectured that row lengths are given by A237271.

			Triangle begins:
   1;
   2;
   1,  3;
   8;
   1,  5;
  36;
   1,  7;
  64;
   1,  3,  9;
   2, 50;
  ...
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] and [5, 10]. The product of terms are 1*2 = 2 and 5*10 = 50 respectively, so the row 10 of the triangle is [2, 50].

Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Row products give A007955.

Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984.

A386989row[n_] :=Times @@@ Split[Divisors[n], #2/# <= 2 &];
Array[A386989row, 50] (* Paolo Xausa, Aug 29 2025 *)

A196149 Numbers whose divisors increase by a factor of 3 or less.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 63, 64, 66, 70, 72, 75, 78, 80, 81, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 117, 120, 126, 128, 130, 132, 135, 136, 140, 144, 147, 150

Offset: 1

Reinhard Zumkeller, Sep 28 2011

nonn

The polymath8 project led by Terry Tao refers to these numbers as "3-densely divisible". In general they say that n is y-densely divisible if its divisors increase by a factor of y or less, or equivalently, if for every R with 1 <= R <= n, there is a divisor in the interval [R/y,R]. They use this as a weakening of the condition that n be y-smooth. - David S. Metzler, Jul 02 2013

Let D(x) denote the number of such integers up to x. D(x) has order of magnitude x/log(x) (See Saias 1997). Moreover, we have D(x) = c*x/log(x) + O(x/(log(x))^2), where c = 2.05544... (See Weingartner 2015, 2019). As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 0.486513... - Andreas Weingartner, Jun 25 2021

			14 is not a term because its divisors are 1,2,7,14, and the gap from 2 to 7 is more than a factor of 3. - _N. J. A. Sloane_, Aug 03 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Terence Tao, A Truncated Elementary Selberg Sieve of Pintz. (blog entry defining y-densely divisible)
Terence Tao et al., Polymath8 home page.
Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.

A174973 is a subsequence.

Cf. A027750.

a196149 n = a196149_list !! (n-1)
a196149_list = filter f [1..] where
   f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 3) divs)
                      where divs = a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

dif3[n_]:=Max[#[[2]]/#[[1]]&/@Partition[Divisors[n],2,1]]<=3; Select[ Range[ 200],dif3] (* Harvey P. Dale, Jun 08 2015 *)

is(n)=my(d=divisors(n));for(i=2,#d,if(d[i]>3*d[i-1],return(0)));1 \\ Charles R Greathouse IV, Jul 06 2013

from sympy import divisors
def ok(n):
    d = divisors(n)
    return all(d[i]/d[i-1] <= 3 for i in range(1, len(d)))
print(list(filter(ok, range(1, 151)))) # Michael S. Branicky, Jun 25 2021

a(n) = A052287(n) / 3.

a(n) = C*n*log(n*log(n)) + O(n), where C = 0.486513… (See comments). - Andreas Weingartner, Jun 25 2021

A317306 Powers of 2 and even perfect numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 28, 32, 64, 128, 256, 496, 512, 1024, 2048, 4096, 8128, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33550336, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589869056, 8589934592

Offset: 1

Omar E. Pol, Aug 23 2018

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).

			Illustration of initial terms:
.        _ _   _   _   _               _                       _       _
.    1  |_| | | | | | | |             | |                     | |     | |
.    2  |_ _|_| | | | | |             | |                     | |     | |
.        _ _|  _|_| | | |             | |                     | |     | |
.    4  |_ _ _|    _|_| |             | |                     | |     | |
.        _ _ _|  _|  _ _|             | |                     | |     | |
.    6  |_ _ _ _|  _|                 | |                     | |     | |
.        _ _ _ _| |                   | |                     | |     | |
.    8  |_ _ _ _ _|              _ _ _| |                     | |     | |
.                               |  _ _ _|                     | |     | |
.                              _| |                           | |     | |
.                            _|  _|                           | |     | |
.                        _ _|  _|                             | |     | |
.                       |  _ _|                               | |     | |
.                       | |                          _ _ _ _ _| |     | |
.        _ _ _ _ _ _ _ _| |                         |  _ _ _ _ _|     | |
.   16  |_ _ _ _ _ _ _ _ _|                         | |    _ _ _ _ _ _| |
.                                                _ _| |   |  _ _ _ _ _ _|
.                                            _ _|  _ _|   | |
.                                           |    _|    _ _| |
.                                          _|  _|     |  _ _|
.                                         |  _|      _| |
.                                    _ _ _| |      _|  _|
.                                   |  _ _ _|  _ _|  _|
.                                   | |       |  _ _|
.                                   | |  _ _ _| |
.                                   | | |  _ _ _|
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.   28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                       | |
.                                       | |
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.   32  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram shows the first eight terms of the sequence. The symmetric representation of sigma has only one part, and apart from the central width, the rest of the widths are 1's.
A317307(n) is the area (or the number of cells) in the n-th region of the diagram.

Union of A000079 and A000396 assuming there are no odd perfect numbers.
Subsequence of A174973.
Cf. A249351 (the widths).
Cf. A317307(n) = sigma(a(n)).
Cf. A000203, A000225, A139256, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626.

cp's OEIS Frontend

A323648 Numbers k such that the largest Dyck path of the symmetric representation of sigma(k) does not share any line segment with the largest Dyck path of the symmetric representation of sigma(k+1).

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A347980 a(n) is the smallest odd number k whose symmetric representation of sigma(k) has maximum width n.

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A348705 a(n) is the total length of all line segments in the symmetric representation of sigma(n).

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A387030 Irregular triangle read by rows: T(n,k) is the number of primes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.

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A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

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A363122 Numbers k such that the highest power of 2 dividing k is larger than the highest power of p dividing k for any odd prime p.

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A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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A196149 Numbers whose divisors increase by a factor of 3 or less.

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