A327329 -id:A327329 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A243980 Four times the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

4, 16, 32, 60, 84, 132, 164, 224, 276, 348, 396, 508, 564, 660, 756, 880, 952, 1108, 1188, 1356, 1484, 1628, 1724, 1964, 2088, 2256, 2416, 2640, 2760, 3048, 3176, 3428, 3620, 3836, 4028, 4392, 4544, 4784, 5008, 5368, 5536, 5920, 6096, 6432, 6744, 7032, 7224, 7720

Offset: 1

Omar E. Pol, Jun 18 2014

nonn

Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.

			Illustration of the structure after 16 stages (contains 880 ON cells):
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Indranil Ghosh, Table of n, a(n) for n = 1..7342

Partial sums of A239050.
Partial sums give A244050.
Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.

Accumulate[4*DivisorSigma[1,Range[50]]] (* Harvey P. Dale, May 13 2018 *)

from math import isqrt
def A243980(n): return -(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))<<1 # Chai Wah Wu, Oct 21 2023

a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).

a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - Omar E. Pol, Sep 25 2019

A340424 Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 4, 0, 8, 0, 1, 15, 0, 4, 1, 21, 0, 8, 4, 2, 33, 0, 15, 8, 8, 2, 41, 0, 21, 15, 16, 8, 4, 56, 0, 33, 21, 30, 16, 16, 4, 69, 0, 41, 33, 42, 30, 32, 16, 7, 87, 0, 56, 41, 66, 42, 60, 32, 28, 8, 99, 0, 69, 56, 82, 66, 84, 60, 56, 32, 12, 127, 0, 87, 69, 112, 82, 132, 84, 105, 64, 48, 14

Offset: 1

Omar E. Pol, Jan 07 2021

Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.

			Triangle begins:
   1;
   4,  0;
   8,  0,  1;
  15,  0,  4,  1;
  21,  0,  8,  4,  2;
  33,  0, 15,  8,  8,  2;
  41,  0, 21, 15, 16   8,  4;
  56,  0, 33, 21, 30, 16, 16,  4;
  69,  0, 41, 33, 42, 30, 32, 16,  7;
  87,  0, 56, 41, 66, 42, 60, 32, 28,  8;
  99,  0, 69, 56, 82, 66, 84, 60, 56, 32, 12;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *  33   =  33
2      0   *  21   =   0
3      1   *  15   =  15
4      1   *   8   =   8
5      2   *   4   =   8
6      2   *   1   =   2
.           A024916
--------------------------
The sum of row 6 is 33 + 0 + 15 + 8 + 8 + 2 = 66, equaling A066186(6) = 66.

Mirror of A245099.
Columns 1, 3 and 4 are A024916 (partial sums of A000203).
Column 2 gives A000004.
Columns 5 and 6 give A327329.
Columns 7 and 8 give A243980.
Leading diagonal gives A002865.
Cf. A066186.

A345023 a(n) is the surface area of the symmetric tower described in A221529 which is a polycube whose successive terraces are the symmetric representation of sigma A000203(i) (from i = 1 to n) starting from the top and the levels of these terraces are the partition numbers A000041(h-1) (from h = 1 to n) starting from the base.

Original entry on oeis.org

6, 16, 32, 58, 90, 142, 202, 292, 406, 562, 754, 1034, 1370, 1822, 2410, 3176, 4136, 5402, 6982, 9026, 11598, 14838, 18894, 24034, 30396, 38312, 48136, 60288, 75220, 93624, 116104, 143598, 177090, 217770, 267106, 326820, 398804, 485472, 589644, 714564, 864000, 1042524, 1255308

Offset: 1

Omar E. Pol, Jun 05 2021

nonn

The largest side of the base of the tower has length n.
The base of the tower is the symmetric representation of A024916(n).
The volume of the tower is equal to A066186(n).
The area of each lateral view of the tower is equal to A000070(n-1).
The growth of the volume of the tower represents the convolution of A000203 and A000041.
The above results are because the correspondence between divisors and partitions described in A338156 and A336812.
The tower is also a member of the family of the stepped pyramid described in A245092.
The equivalent sequence for the surface area of the stepped pyramid is A328366.

			For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1).
     _ _ _ _ _ _ _
    |_ _ _ _      |                 7
    |_ _ _ _|_    |           4     3
    |_ _ _    |   |             5   2
    |_ _ _|_ _|_  |         3   2   2                                    _
    |_ _ _      | |               6 1                 1                 | |
    |_ _ _|_    | |         3     3 1                 1                 | |
    |_ _    |   | |           4   2 1                 1                 | |
    |_ _|_ _|_  | |       2   2   2 1                 1                _|_|
    |_ _ _    | | |             5 1 1               1 1               |   |
    |_ _ _|_  | | |         3   2 1 1               1 1              _|_ _|
    |_ _    | | | |           4 1 1 1             1 1 1             | |   |
    |_ _|_  | | | |       2   2 1 1 1             1 1 1            _|_|_ _|
    |_ _  | | | | |         3 1 1 1 1           1 1 1 1          _| |_ _ _|
    |_  | | | | | |       2 1 1 1 1 1         1 1 1 1 1      _ _|_ _|_ _ _|
    |_|_|_|_|_|_|_|     1 1 1 1 1 1 1     1 1 1 1 1 1 1     |_ _|_|_ _ _ _|
.
       Figure 1.           Figure 2.        Figure 3.           Figure 4.
   Front view of the      Partitions        Position          Lateral view
  prism of partitions.       of 7.         of the 1's.        of the tower.
.
.
                                                             _ _ _ _ _ _ _
                                                            |   | | | | |_|  1
                                                            |   | | |_|_ _|  2
                                                            |   |_|_  |_ _|  3
                                                            |_ _    |_ _ _|  4
                                                                |_  |_ _ _|  5
                                                                  |       |  6
                                                                  |_ _ _ _|  7
.
                                                               Figure 5.
                                                               Top view
                                                             of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 7. The area of the diagram is A066186(7) = 105. Note that the diagram can be interpreted also as the front view of a right prism whose volumen is 1*7*A000041(7) = 1*7*15 = 105, equaling the volume of the tower that appears in the figures 4 and 5.
Figure 2 shows the partitions of 7 in accordance with the diagram.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions, see the figures 3 and 4. In this case the mentioned area equals A000070(7-1) = 30.
The connection between these two objects is a representation of the correspondence divisor/part described in A338156. See also A336812.

Cf. A000041, A000070, A000203, A024916, A066186, A176206, A196020, A221529, A236104, A237593, A244050, A245092, A327329, A328366, A336811, A336812, A338156.

Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k], {k, 1, 50}] (* Amiram Eldar, Jul 14 2021 *)

a(n) = 4*A000070(n-1) + 2*A024916(n).

a(n) = 4*A000070(n-1) + A327329(n).

A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.

Original entry on oeis.org

0, 0, 2, 3, 6, 9, 16, 13, 27, 31, 34, 43, 57, 56, 75, 80, 96, 99, 121, 122, 155, 164, 163, 184, 220, 218, 255, 252, 277, 304, 339, 328, 372, 389, 412, 433, 491, 478, 515, 536, 570, 609, 638, 647, 722, 713, 746, 767, 858, 842, 910, 939, 942, 993, 1060, 1057

Offset: 1

Amarnath Murthy, Jan 29 2002

			a(7) = 16 = 1 +3 +6 +4 +2 = 13 % 3 + 13 % 5 + 13 % 7 + 13 % 9 + 13 % 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A000593, A004125, A067436, A024916, A327329.

L:= [seq(4*n-3 - numtheory:-sigma(2*n-1)-numtheory:-sigma((n-1)/2^padic:-ordp(n-1,2)), n=1..100)]:
ListTools:-PartialSums(L); # Robert Israel, Jan 16 2019

from math import isqrt
def A327329(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))
def A067435(n): return n*((n<<1)-1)-(A327329(n<<1)>>1)-A327329(n>>1)+3*(A327329(n)>>1)+A327329(n-1>>1)-(A327329(n-1)>>1) # Chai Wah Wu, Nov 01 2023

a(n) = a(n-1) + 4*n-3 - A000203(2*n-1) - A000593(n-1). - Robert Israel, Jan 16 2019

a(n) = n*(2*n-1) - A326123(n) - A078471(n-1) = n*(2*n-1) - A024916(2*n) - 2*A024916(floor(n/2)) + 3*A024916(n) + 2*A024916(floor((n-1)/2)) - A024916(n-1). - Chai Wah Wu, Nov 01 2023

Corrected and extended by several contributors.

A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 8, 4, 2, 15, 8, 8, 3, 21, 15, 16, 12, 5, 33, 21, 30, 24, 20, 7, 41, 33, 42, 45, 40, 28, 11, 56, 41, 66, 63, 75, 56, 44, 15, 69, 56, 82, 99, 105, 105, 88, 60, 22, 87, 69, 112, 123, 165, 147, 165, 120, 88, 30, 99, 87, 138, 168, 205, 231, 231, 225, 176, 120, 42, 127, 99, 174

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture 1: T(n,k) is the sum of divisors of the terms that are in the k-th blocks of the first n rows of triangle A176206.
Conjecture 2: the sum of row n equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
Conjecture 3: T(n,k) is also the volume (or number of cubes) of the k-th block of a symmetric tower in which the terraces are the symmetric representation of sigma (n..1) starting from the base respectively (cf. A237270, A237593), hence the total area of the terraces is A024916(n), the same as the area of the base.
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1). Hence the differences between levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
This symmetric tower has the property that its volume (or total number of cubes) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
For another symmetric tower of the same family and whose volume equals A066186(n) see A339106 and A221529.
The above three conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).

			Triangle begins:
   1;
   4,   1;
   8,   4,   2;
  15,   8,   8,   3;
  21,  15,  16,  12,   5;
  33,  21,  30,  24,  20,   7;
  41,  33,  42,  45,  40,  28,  11;
  56,  41,  66,  63,  75,  56,  44,  15;
  69,  56,  82,  99, 105, 105,  88,  60,  22;
  87,  69, 112, 123, 165, 147, 165, 120,  88,  30;
  99,  87, 138, 168, 205, 231, 231, 225, 176, 120,  42;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000041         T(6,k)
1      1  *  33   =   33
2      1  *  21   =   21
3      2  *  15   =   30
4      3  *   8   =   24
5      5  *   4   =   20
6      7  *   1   =    7
.          A024916
--------------------------
The sum of row 6 is 33 + 21 + 30 + 24 + 20 + 7 = 135, equaling A182738(6).

Columns 1 and 2 give A024916.
Column 3 gives A327329.
Leading diagonal gives A000041.
Row sums give A182738.
Cf. A000070, A066186, A176206, A221529, A221531, A237270, A237593, A336811, A336812, A338156, A339106, A340035,  A340424, A340425, A340426, A340524, A340526.

A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 20, 40, 70, 102, 150, 194, 256, 318, 394, 462, 566, 646, 750, 858, 984, 1088, 1238, 1354, 1518, 1666, 1826, 1966, 2182, 2344, 2532, 2720, 2944, 3120, 3384, 3572, 3826, 4054, 4298, 4534, 4860, 5084, 5356, 5624, 5964, 6212, 6572, 6832, 7176, 7512, 7840, 8124, 8564, 8874, 9260, 9608, 10012

Offset: 1

Omar E. Pol, Oct 26 2019

nonn

			For n = 1 the first level of the stepped pyramid is a cube, so a(1) = 6.

Cf. A000217, A002378, A013661, A024916, A045943, A046092, A153485, A175254 (volume), A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A327329.

s=0;Do[s=s+4*DivisorSigma[1,n];t=2n(n+1);Print[(s/2)+t],{n,1,8000}] (* Metin Sariyar, Nov 20 2019 *)

from math import isqrt
def A328366(n): return (n*(n+1)<<1)-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 22 2023

a(n) = 4*A000217(n) + 2*A024916(n).
a(n) = 2*(A002378(n) + A327329(n)).
a(n) = 2*(A045943(n) + A153485(n)).
a(n) = A046092(n) + A327329(n).
a(n) = 2*A299692(n).
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2) + 2 = 3.644934... . - Amiram Eldar, Mar 21 2024

cp's OEIS Frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

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A243980 Four times the sum of all divisors of all positive integers <= n.

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A340424 Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

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A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.

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A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

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A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

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