A380580 -id:A380580 - cp's OEIS frontend

A237271 Number of parts in the symmetric representation of sigma(n).

1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 1, 3, 2, 4, 1, 2, 1, 2, 1, 4, 2, 3, 1, 2, 2, 4, 1, 2, 1, 2, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 4, 1, 4, 2, 2, 1, 2, 2, 5, 1, 4, 1, 2, 2, 4, 3, 2, 1, 2, 2, 4, 2, 3, 2, 2, 1, 5, 2, 2, 1, 4, 2, 4, 1, 2, 1

Offset: 1

Omar E. Pol, Feb 25 2014

nonn

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

			Illustration of initial terms (n = 1..12):
---------------------------------------------------------
n   A000203  A237270    a(n)            Diagram
---------------------------------------------------------
.                               _ _ _ _ _ _ _ _ _ _ _ _
1       1      1         1     |_| | | | | | | | | | | |
2       3      3         1     |_ _|_| | | | | | | | | |
3       4      2+2       2     |_ _|  _|_| | | | | | | |
4       7      7         1     |_ _ _|    _|_| | | | | |
5       6      3+3       2     |_ _ _|  _|  _ _|_| | | |
6      12      12        1     |_ _ _ _|  _| |  _ _|_| |
7       8      4+4       2     |_ _ _ _| |_ _|_|    _ _|
8      15      15        1     |_ _ _ _ _|  _|     |
9      13      5+3+5     3     |_ _ _ _ _| |      _|
10     18      9+9       2     |_ _ _ _ _ _|  _ _|
11     12      6+6       2     |_ _ _ _ _ _| |
12     28      28        1     |_ _ _ _ _ _ _|
...
For n = 9 the sum of divisors of 9 is 1+3+9 = A000203(9) = 13. On the other hand the 9th set of symmetric regions of the diagram is formed by three regions (or parts) with 5, 3 and 5 cells, so the total number of cells is 5+3+5 = 13, equaling the sum of divisors of 9. There are three parts: [5, 3, 5], so a(9) = 3.
From _Omar E. Pol_, Dec 21 2016: (Start)
Illustration of the diagram of subparts (n = 1..12):
---------------------------------------------------------
n   A000203  A279391  A001227           Diagram
---------------------------------------------------------
.                               _ _ _ _ _ _ _ _ _ _ _ _
1       1      1         1     |_| | | | | | | | | | | |
2       3      3         1     |_ _|_| | | | | | | | | |
3       4      2+2       2     |_ _|  _|_| | | | | | | |
4       7      7         1     |_ _ _|  _ _|_| | | | | |
5       6      3+3       2     |_ _ _| |_|  _ _|_| | | |
6      12      11+1      2     |_ _ _ _|  _| |  _ _|_| |
7       8      4+4       2     |_ _ _ _| |_ _|_|  _ _ _|
8      15      15        1     |_ _ _ _ _|  _|  _| |
9      13      5+3+5     3     |_ _ _ _ _| |  _|  _|
10     18      9+9       2     |_ _ _ _ _ _| |_ _|
11     12      6+6       2     |_ _ _ _ _ _| |
12     28      23+5      2     |_ _ _ _ _ _ _|
...
For n = 6 the symmetric representation of sigma(6) has two subparts: [11, 1], so A000203(6) = 12 and A001227(6) = 2.
For n = 12 the symmetric representation of sigma(12) has two subparts: [23, 5], so A000203(12) = 28 and A001227(12) = 2. (End)
From _Hartmut F. W. Hoft_, Dec 26 2016: (Start)
Two examples of the general argument in the Comments section:
Rows 27 in A237048 and A249223 (4 parts)
i:  1  2 3 4 5 6 7 8 9 . . 12
27: 1  1 1 0 0 1                           1's in A237048 for odd divisors
    1 27 3     9                           odd divisors represented
27: 1  0 1 1 1 0 0 1 1 1 0 1               blocks forming parts in A249223
Rows 81 in A237048 and A249223 (5 parts)
i:  1  2 3 4 5 6 7 8 9 . . 12. . . 16. . . 20. . . 24
81: 1  1 1 0 0 1 0 0 1 0 0 0                          1's in A237048 f.o.d
    1 81 3    27     9                                odd div. represented
81: 1  0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1  blocks fp in A249223
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michel Marcus)

Row lengths of A237270 and of A379288.
Column 1 of A279387.
Partial sums give A237590.
Parity gives A347950.
Cf. A000203, A000265, A001065, A001227, A005279, A024916, A060831, A061345, A067742, A071561, A071562, A175254, A196020, A221529, A235791, A236104, A237048, A237591, A237593, A239657, A244050, A244971, A245092, A249223, A250068, A261699, A262045, A262612, A262626, A274824, A279387, A279693, A319073, A340583, A340846, A342344, A347186, A379288.
Cf. A027750, A174973 (2-dense numbers), A239663, A240062, A243982, A379379, A380580, A384149, A384222, A384225, A384226, A384230, A384930.

a237271[n_] := Length[a237270[n]] (* code defined in A237270 *)
Map[a237271, Range[90]] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)
a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, 1 + Count[d, ?(OddQ[#[[2]]] && #[[2]] >= 2*#[[1]] &)]]; Array[a, 100] (* _Amiram Eldar,  Dec 22 2024 *)

fill(vcells, hga, hgb) = {ic = 1; for (i=1, #hgb, if (hga[i] < hgb[i], for (j=hga[i], hgb[i]-1, cell = vector(4); cell[1] = i - 1; cell[2] = j; vcells[ic] = cell; ic ++;););); vcells;}
findfree(vcells) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[3] == 0) && (vcelli[4] == 0), return (i));); return (0);}
findxy(vcells, x, y) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[1]==x) && (vcelli[2]==y) && (vcelli[3] == 0) && (vcelli[4] == 0), return (i));); return (0);}
findtodo(vcells, iz) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[3] == iz) && (vcelli[4] == 0), return (i)); ); return (0);}
zcount(vcells) = {nbz = 0; for (i=1, #vcells, nbz = max(nbz, vcells[i][3]);); nbz;}
docell(vcells, ic, iz) = {x = vcells[ic][1]; y = vcells[ic][2]; if (icdo = findxy(vcells, x-1, y), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x+1, y), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x, y-1), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x, y+1), vcells[icdo][3] = iz); vcells[ic][4] = 1; vcells;}
docells(vcells, ic, iz) = {vcells[ic][3] = iz; while (ic, vcells = docell(vcells, ic, iz); ic = findtodo(vcells, iz);); vcells;}
nbzb(n, hga, hgb) = {vcells = vector(sigma(n)); vcells = fill(vcells, hga, hgb); iz = 1; while (ic = findfree(vcells), vcells = docells(vcells, ic, iz); iz++;); zcount(vcells);}
lista(nn) = {hga = concat(heights(row237593(0), 0), 0); for (n=1, nn, hgb = heights(row237593(n), n); nbz = nbzb(n, hga, hgb); print1(nbz, ", "); hga = concat(hgb, 0););} \\ with heights() also defined in A237593; \\ Michel Marcus, Mar 28 2014

from sympy import divisors
def a(n: int) -> int:
    divs = list(divisors(n))
    d = [divs[i:i+2] for i in range(len(divs) - 1)]
    s = sum(1 for pair in d if len(pair) == 2 and pair[1] % 2 == 1 and pair[1] >= 2 * pair[0])
    return s + 1
print([a(n) for n in range(1, 80)])  # Peter Luschny, Aug 05 2025

a(n) = A001227(n) - A239657(n). - Omar E. Pol, Mar 23 2014
a(p^k) = k + 1, where p is an odd prime and k >= 0. - Hartmut F. W. Hoft, Dec 26 2016
Theorem: a(n) <= number of odd divisors of n (cf. A001227). The differences are in A239657. - N. J. A. Sloane, Jan 19 2021
a(n) = A340846(n) - A340833(n) + 1 (Euler's formula). - Omar E. Pol, Feb 01 2021
a(n) = A000005(n) - A243982(n). - Omar E. Pol, Aug 02 2025

A380579 Triangle read by rows in which row n lists n successive integers in descending order starting with the n-th positive integer not divisible by 3, with n >= 1 and 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 5, 4, 3, 2, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 3, 10, 9, 8, 7, 6, 5, 4, 11, 10, 9, 8, 7, 6, 5, 4, 13, 12, 11, 10, 9, 8, 7, 6, 5, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7

Offset: 1

Omar E. Pol, Jan 29 2025

This sequence is mentioned in the Name section of A380580. That sequence represents a template for a Pop-Up pyramid which is related to Combinatorics, Geometry, Number Theory and several tens of integers sequences.
The n-th row of this triangle can be visualized in the template n - 1.
The sum of the n-th row equals the area of the largest polygon in the template n - 1.
In this triangle the last term of the row n is equal to both A237591(n-1,1) and A237593(n-1,1).
The m-th diagonal lists the terms of A008619 but starting from the term whose index is 3*m - 3, with m >= 1.
The column 3*m - 2 lists the terms of A001651 but starting from the m-th term, m >= 1.
The column 3*m - 1 lists the terms of A032766 but starting from the m-th term, m >= 1.
The column 3*m lists the terms of A007494 but starting from the m-th term, m >= 1.

			Triangle begins:
   1;
   2,  1;
   4,  3,  2;
   5,  4,  3,  2;
   7,  6,  5,  4,  3;
   8,  7,  6,  5,  4,  3;
  10,  9,  8,  7,  6,  5,  4;
  11, 10,  9,  8,  7,  6,  5,  4;
  13, 12, 11, 10,  9,  8,  7,  6,  5;
  14, 13, 12, 11, 10,  9,  8,  7,  6,  5;
  16, 15, 14, 13, 12, 11, 10,  9,  8,  7,  6;
  17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7,  6;
  19, 18, 17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7;
  20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7;
  ...
For n = 5 the illustration of the row 5 of the triangle as the column 1 and also as the right border of the 4th slice of A380580 is as shown below:
              _ _ _ _ _ _ _ _ _ _ _ _ _ _
         7   |            _|_            |   7
         6   |          _|_|_|_          |   6
         5   |        _|_ _|_ _|_        |   5
         4   |      _|_ _|_|_|_ _|_      |   4
         3   |_ _ _|_ _ _|_|_|_ _ _|_ _ _|   3
.
The last term of the row 5 is equal to 3, the same as both A237591(4,1) = 3 and A237593(4,1) = 3.
The sum of the 5th row of this triangle is 7 + 6 + 5 + 4 + 3 = 25, the same as the area of largest polygon of the diagram.
.
For n = 6 the illustration of the row 6 of the triangle as the column 1 and also as the right border of the 5th slice of A380580 is as shown below:
            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
       8   |              _|_              |   8
       7   |            _|_|_|_            |   7
       6   |          _|_ _|_ _|_          |   6
       5   |        _|_ _|_|_|_ _|_        |   5
       4   |      _|_ _ _|_|_|_ _ _|_      |   4
       3   |_ _ _|_ _ _|_ _|_ _|_ _ _|_ _ _|   3
.
The last term of the row 6 is equal to 3, the same as both A237591(5,1) = 3 and A237593(5,1) = 3.
The sum of the 6th row of this triangle is 8 + 7 + 6 + 5 + 4 + 3 = 33, the same as the area of the largest polygon of the diagram.
.
For n = 7 the illustration of the row 7 of the triangle as the column 1 and also as the right border of the 6th slice of A380580 is as shown below:
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
  10   |                  _|_                  |   10
   9   |                _|_|_|_                |    9
   8   |              _|_ _|_ _|_              |    8
   7   |            _|_ _|_|_|_ _|_            |    7
   6   |          _|_ _ _|_|_|_ _ _|_          |    6
   5   |        _|_ _ _|_ _|_ _|_ _ _|_        |    5
   4   |_ _ _ _|_ _ _ _|_|_|_|_|_ _ _ _|_ _ _ _|    4
.
The last term of the row 7 is equal to 4, the same as both A237591(6,1) = 4 and A237593(6,1) = 4.
The sum of the 7th row of this triangle is 10 + 9 + 8 + 7 + 6 + 5 + 4 = 49, the same as the area of the largest polygon of the diagram.
.

Companion of A380580.
Subsequence of A004736.
Column 1 gives A001651.
Column 2 gives the nonzero terms of A032766.
Column 3 gives the nonzero terms of A007494.
Middle diagonal gives A005408.
Leading diagonal gives A008619.
Cf. A237591, A237593.

T[n_,k_]:=Floor[(3*n-1)/2]-k+1; Table[T[n,k],{n,13},{k,n}]//Flatten (* Stefano Spezia, Apr 24 2025 *)

T(n,k) = A001651(n) - k + 1.

G.f.: x*y*(1 + x - x^4*y^2 + x^2*(1 + y) - x^3*y*(1 + 2*y))/((1 - x)^2*(1 + x)*(1 - x*y)^2*(1 + x*y)). - Stefano Spezia, Apr 24 2025

cp's OEIS Frontend

A237271 Number of parts in the symmetric representation of sigma(n).

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