A224880 -id:A224880 - cp's OEIS frontend

A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved.

0, 1, 2, 3, 4, 4, 6, 7, 8, 6, 10, 12, 12, 8, 14, 15, 16, 13, 18, 18, 20, 12, 22, 28, 24, 14, 26, 24, 28, 24, 30, 31, 32, 18, 34, 39, 36, 20, 38, 42, 40, 32, 42, 36, 44, 24, 46, 60, 48, 31, 50, 42, 52, 40, 54, 56, 56, 30, 58, 72, 60, 32, 62, 63, 64, 48

Offset: 0

Omar E. Pol, Jul 15 2014

nonn

Consider an irregular stepped pyramid with n steps. The base of the pyramid is equal to the symmetric representation of A024916(n), the sum of all divisors of all positive integers <= n. Two of the faces of the pyramid are the same as the representation of the n-th triangular numbers as a staircase. The total area of the pyramid is equal to 2*A024916(n) + A046092(n). The volume is equal to A175254(n). By definition a(2n-1) is A000203(n), the sum of divisors of n. Starting from the top a(2n-1) is also the total area of the horizontal part of the n-th step of the pyramid. By definition, a(2n) = A005843(n) = 2n. Starting from the top, a(2n) is also the total area of the irregular vertical part of the n-th step of the pyramid.
On the other hand the sequence also has a symmetric representation in two dimensions, see Example.
From Omar E. Pol, Dec 31 2016: (Start)
We can find the pyramid after the following sequences: A196020 --> A236104 --> A235791 --> A237591 --> A237593.
The structure of this infinite pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see the links).
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1, hence the sum of the areas of the terraces at the m-th level equals A000203(m).
Note that the stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the pyramid see A237593 and all its related sequences. (End)

			Illustration of initial terms:
----------------------------------------------------------------------
a(n)                             Diagram
----------------------------------------------------------------------
0    _
1   |_|\ _
2    \ _| |\ _
3     |_ _| | |\ _
4      \ _ _|_| | |\ _
4       |_ _|  _| | | |\ _
6        \ _ _|  _| | | | |\ _
7         |_ _ _|  _|_| | | | |\ _
8          \ _ _ _|  _ _| | | | | |\ _
6           |_ _ _| |    _| | | | | | |\ _
10           \ _ _ _|  _|  _|_| | | | | | |\ _
12            |_ _ _ _|  _|  _ _| | | | | | | |\ _
12             \ _ _ _ _|  _|  _ _| | | | | | | | |\ _
8               |_ _ _ _| |  _|  _ _|_| | | | | | | | |\ _
14               \ _ _ _ _| |  _| |  _ _| | | | | | | | | |\ _
15                |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |\ _
16                 \ _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |\
13                  |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | | |
18                   \ _ _ _ _ _| |  _|  _|    _ _| | | | | | | | | |
18                    |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | | |
20                     \ _ _ _ _ _ _| |      _| |  _ _ _| | | | | | |
12                      |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | | |
22                       \ _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | | |
28                        |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| | |
24                         \ _ _ _ _ _ _ _| |  _| |    _| |  _ _ _| |
14                          |_ _ _ _ _ _ _| | |  _|  _|  _| |  _ _ _|
26                           \ _ _ _ _ _ _ _| | |_ _|  _|  _| |
24                            |_ _ _ _ _ _ _ _| |  _ _|  _|  _|
28                             \ _ _ _ _ _ _ _ _| |  _ _|  _|
24                              |_ _ _ _ _ _ _ _| | |  _ _|
30                               \ _ _ _ _ _ _ _ _| | |
31                                |_ _ _ _ _ _ _ _ _| |
32                                 \ _ _ _ _ _ _ _ _ _|
...
a(n) is the total area of the n-th set of symmetric regions in the diagram.
.
From _Omar E. Pol_, Aug 21 2015: (Start)
The above structure contains a hidden pattern, simpler, as shown below:
Level                              _ _
1                                _| | |_
2                              _|  _|_  |_
3                            _|   | | |   |_
4                          _|    _| | |_    |_
5                        _|     |  _|_  |     |_
6                      _|      _| | | | |_      |_
7                    _|       |   | | |   |       |_
8                  _|        _|  _| | |_  |_        |_
9                _|         |   |  _|_  |   |         |_
10             _|          _|   | | | | |   |_          |_
11           _|           |    _| | | | |_    |           |_
12         _|            _|   |   | | |   |   |_            |_
13       _|             |     |  _| | |_  |     |             |_
14     _|              _|    _| |  _|_  | |_    |_              |_
15   _|               |     |   | | | | |   |     |               |_
16  |                 |     |   | | | | |   |     |                 |
...
The symmetric pattern emerges from the front view of the stepped pyramid.
Note that starting from this diagram A000203 is obtained as follows:
In the pyramid the area of the k-th vertical region in the n-th level on the front view is equal to A237593(n,k), and the sum of all areas of the vertical regions in the n-th level on the front view is equal to 2n.
The area of the k-th horizontal region in the n-th level is equal to A237270(n,k), and the sum of all areas of the horizontal regions in the n-th level is equal to sigma(n) = A000203(n). (End)
From _Omar E. Pol_, Dec 31 2016: (Start)
Illustration of the top view of the pyramid with 16 levels:
.
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      |_ _ _ _ _ _ _ _ _|
... (End)

Robert Price, Table of n, a(n) for n = 0..20000
Omar E. Pol, A pyramid related to the divisor function and other integers sequences
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 pyramid (first 16 levels)

Cf. A000203, A004125, A024916, A005843, A175254, A196020, A224880, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244970, A244971, A245093, A261350, A262626, A277437, A279387, A280223, A280295.

Table[If[EvenQ@ n, n, DivisorSigma[1, (n + 1)/2]], {n, 0, 65}] (* or *)
Transpose@ {Range[0, #, 2], DivisorSigma[1, #] & /@ Range[#/2 + 1]} &@ 65 // Flatten (* Michael De Vlieger, Dec 31 2016 *)
With[{nn=70},Riffle[Range[0,nn,2],DivisorSigma[1,Range[nn/2]]]] (* Harvey P. Dale, Aug 05 2024 *)

a(2*n-1) + a(2n) = A224880(n).

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.

Original entry on oeis.org

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.

A318468 a(n) = 2*n AND A000203(n), where AND is bitwise-and (A004198) and A000203 = sum of divisors.

Original entry on oeis.org

0, 0, 4, 0, 2, 12, 8, 0, 0, 16, 4, 24, 10, 24, 24, 0, 2, 36, 4, 40, 32, 36, 8, 48, 18, 32, 32, 56, 26, 8, 32, 0, 0, 4, 0, 72, 2, 12, 8, 80, 2, 64, 4, 80, 74, 72, 16, 96, 32, 68, 64, 96, 34, 104, 72, 112, 80, 80, 52, 40, 58, 96, 104, 0, 0, 128, 4, 8, 0, 128, 8, 128, 2, 16, 20, 136, 0, 136, 16, 160, 32, 36, 4, 160, 40, 132, 40, 176, 18, 160, 48

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003987, A318466, A318467.

Cf. also A318458.

Array[BitAnd[2 #, DivisorSigma[1, #]] &, 91] (* Michael De Vlieger, Apr 21 2019 *)

PARI
```
A318468(n) = bitand(2*n,sigma(n));
```

a(n) = A004198(2*n, A000203(n)).

a(n) = A224880(n) - A318466(n) = (A224880(n)-A318467(n))/2.

A318466 a(n) = 2*n OR A000203(n), where OR is bitwise-or (A003986) and A000203 = sum of divisors.

Original entry on oeis.org

3, 7, 6, 15, 14, 12, 14, 31, 31, 22, 30, 28, 30, 28, 30, 63, 50, 39, 54, 42, 42, 44, 62, 60, 63, 62, 62, 56, 62, 124, 62, 127, 114, 118, 118, 91, 110, 124, 126, 90, 122, 116, 126, 92, 94, 92, 126, 124, 123, 125, 110, 106, 126, 124, 110, 120, 114, 126, 126, 248, 126, 124, 126, 255, 214, 148, 198, 254, 234, 156, 206

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003986, A224880, A318467, A318468.

Cf. also A318456.

Array[BitOr[2 #, DivisorSigma[1, #]] &, 71] (* Michael De Vlieger, Mar 30 2019 *)

PARI
```
A318466(n) = bitor(2*n,sigma(n));
```

from sympy import divisor_sigma
def A318466(n): return (n<<1)|int(divisor_sigma(n)) # Chai Wah Wu, Jul 10 2022

a(n) = A003986(2*n, A000203(n)).

a(n) = A224880(n) - A318468(n).

A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

Original entry on oeis.org

3, 7, 2, 15, 12, 0, 6, 31, 31, 6, 26, 4, 20, 4, 6, 63, 48, 3, 50, 2, 10, 8, 54, 12, 45, 30, 30, 0, 36, 116, 30, 127, 114, 114, 118, 19, 108, 112, 118, 10, 120, 52, 122, 12, 20, 20, 110, 28, 91, 57, 46, 10, 92, 20, 38, 8, 34, 46, 74, 208, 68, 28, 22, 255, 214, 20, 194, 246, 234, 28, 198, 83, 216, 230, 234, 20, 250, 52, 206, 26

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003987, A318457, A318458, A318466, A318468.
Cf. A000396 (positions of zeros), A378227 (XOR-Moebius transform), A379234 (fixed points), A379236.
Cf. also A294899, A318457, A378988.

Table[BitXor[2n,DivisorSigma[1,n]],{n,80}] (* Harvey P. Dale, Oct 30 2022 *)

PARI
```
A318467(n) = bitxor(2*n,sigma(n));
```

a(n) = A003987(2*n, A000203(n)).
a(n) = A224880(n) - 2*A318468(n).
a(n) = 2*n XOR (A318457(n)+2*A318458(n)). - Antti Karttunen, Jan 08 2025

A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 10, 20, 35, 51, 75, 97, 128, 159, 197, 231, 283, 323, 375, 429, 492, 544, 619, 677, 759, 833, 913, 983, 1091, 1172, 1266, 1360, 1472, 1560, 1692, 1786, 1913, 2027, 2149, 2267, 2430, 2542, 2678, 2812, 2982, 3106, 3286, 3416, 3588, 3756, 3920, 4062, 4282, 4437, 4630, 4804, 5006, 5166, 5394, 5576, 5808, 6002

Offset: 1

Omar E. Pol, Mar 06 2018

nonn

a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).

a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.

			For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.
For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).

Omar E. Pol, Perspective view of the pyramid with 16 levels, contains 492 cells.

Partial sums of A224880.

Cf. A002378, A024916, A045943, A072691, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.

Accumulate[Table[DivisorSigma[1, n] + 2*n, {n, 1, 50}]] (* Amiram Eldar, Mar 21 2024 *)

a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ Michel Marcus, Jun 21 2018

from math import isqrt
def A299692(n): return n*(n+1)+(-(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 22 2023

a(n) = A024916(n) + A002378(n).
a(n) = A153485(n) + A045943(n).
a(n) = A328366(n)/2. - Omar E. Pol, Apr 22 2020
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 21 2024

A339496 T(n, k) = Sum(divisors(k) union {k*j : j = 2..floor(n/k)}). Triangle read by rows.

Original entry on oeis.org

1, 3, 3, 6, 3, 4, 10, 7, 4, 7, 15, 7, 4, 7, 6, 21, 13, 10, 7, 6, 12, 28, 13, 10, 7, 6, 12, 8, 36, 21, 10, 15, 6, 12, 8, 15, 45, 21, 19, 15, 6, 12, 8, 15, 13, 55, 31, 19, 15, 16, 12, 8, 15, 13, 18, 66, 31, 19, 15, 16, 12, 8, 15, 13, 18, 12, 78, 43, 31, 27, 16, 24, 8, 15, 13, 18, 12, 28

Offset: 1

Peter Luschny, Dec 31 2020

For the connection with paths in the divisor graph of {1,...,n} see the comment in A339492.

			The triangle starts:
[1]                       1;
[2]                      3, 3;
[3]                    6, 3, 4;
[4]                  10, 7, 4, 7;
[5]                15, 7, 4, 7, 6;
[6]              21, 13, 10, 7, 6, 12;
[7]            28, 13, 10, 7, 6, 12, 8;
[8]          36, 21, 10, 15, 6, 12, 8, 15;
[9]        45, 21, 19, 15, 6, 12, 8, 15, 13;
[10]     55, 31, 19, 15, 16, 12, 8, 15, 13, 18.

T(n, 1) = A000217(n), T(n, n) = A000203(n), T(2n, n) = A224880(n).

Cf. A339491, A339492, A339489.

t := (n, k) -> NumberTheory:-Divisors(k) union {seq(k*j,j=2..n/k)}:
T := (n, k) -> add(j, j = t(n, k)):
for n from 1 to 10 do seq(T(n, k), k=1..n) od;

A323645 a(n) is the number of visible faces in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 6, 12, 17, 21, 28, 32, 39, 46, 56, 60, 69, 73, 81, 92, 103, 107, 118, 122

Offset: 1

Omar E. Pol, Mar 20 2019

The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).

For more sequences that we can find in the pyramid see A262626.

Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16) and A000203

Cf. A000040, A000203, A024916, A196020, A215848, A224880, A235791, A236104, A237048, A237270, A237271, A237593, A245092, A262626, A299692 (total area).

a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

a(n) = A325300(n) - 3. - Omar E. Pol, Apr 17 2019

a(18)-a(19) from Omar E. Pol, Apr 18 2019

A387001 Number of vertices in the diagram called "symmetric representation of sigma(n)" where its "parts" or polygons are dissected into unit squares (see the example).

Original entry on oeis.org

4, 8, 11, 16, 17, 25, 23, 32, 32, 39, 35, 53, 41, 53, 55, 64, 53, 76, 59, 83, 75, 81, 71, 109, 82, 95, 95, 113, 89, 133, 95, 128, 115, 123, 119, 164, 113, 137, 135, 171, 125, 181, 131, 173, 169, 165, 143, 221, 156, 194, 175, 203, 161, 229, 183, 233, 195, 207, 179, 289, 185, 221, 231, 256

Offset: 1

Omar E. Pol, Aug 14 2025

Consider here that in the diagram every edge has length 1 and every face is a unit square.
The number of faces is A000203(n).
The number of edges is 2*A155085(n).
The number of edges with the same orientation is A155085(n).

			For n = 5 the diagram is as shown below:
   _ _ _
  |_|_|_|
        |_ _
          |_|
          |_|
          |_|
.
The number of vertices is a(5) = 17.
The number of faces is A000203(5) = 6.
The number of edges is 2*A155085(5) = 2*11 = 22.
The number of edges with the same orientation is A155085(5) = 11.

Omar E. Pol, Illustration of initial terms of A000203 in a concave polyhedron (n = 1..16)

Cf. A000203, A005408, A155085, A224880, A237270, A237271, A237593.

a(n) = A000203(n) + A005408(n).
a(n) = 2*A155085(n) - A000203(n) + 1. (Euler's formula: V = E - F + 1).
a(n) = A224880(n) + 1.

cp's OEIS Frontend

A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved.

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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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A318468 a(n) = 2*n AND A000203(n), where AND is bitwise-and (A004198) and A000203 = sum of divisors.

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A318466 a(n) = 2*n OR A000203(n), where OR is bitwise-or (A003986) and A000203 = sum of divisors.

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A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

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A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

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A339496 T(n, k) = Sum(divisors(k) union {k*j : j = 2..floor(n/k)}). Triangle read by rows.

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A323645 a(n) is the number of visible faces in the perspective view of the stepped pyramid with n levels described in A245092.

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