A243980 -id:A243980 - cp's OEIS frontend

A244050 Partial sums of A243980.

4, 20, 52, 112, 196, 328, 492, 716, 992, 1340, 1736, 2244, 2808, 3468, 4224, 5104, 6056, 7164, 8352, 9708, 11192, 12820, 14544, 16508, 18596, 20852, 23268, 25908, 28668, 31716, 34892, 38320, 41940, 45776, 49804, 54196, 58740, 63524, 68532, 73900

Offset: 1

Omar E. Pol, Jun 18 2014

nonn

a(n) is also the volume of a special stepped pyramid with n levels related to the symmetric representation of sigma. Note that starting at the top of the pyramid, the total area of the horizontal regions at the n-th level is equal to A239050(n), and the total area of the vertical regions at the n-th level is equal to 8*n.
From Omar E. Pol, Sep 19 2015: (Start)
Also, consider that the area of the central square in the top of the pyramid is equal to 1, so the total area of the horizontal regions at the n-th level starting from the top is equal to sigma(n) = A000203(n), and the total area of the vertical regions at the n-th level is equal to 2*n.
Also note that this stepped pyramid can be constructed with four copies of the stepped pyramid described in A245092 back-to-back (one copy in every quadrant). (End)
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and the nonzero terms of A008586.
Convolution of A074400 and the nonzero terms of A005843.
Convolution of A340793 and the nonzero terms of A046092.
Convolution of A239050 and A000027.
(End)

			From _Omar E. Pol_, Aug 29 2015: (Start)
Illustration of the top view of the stepped pyramid with 16 levels. The pyramid is formed of 5104 unit cubes:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid.
For more information about the hidden pattern see A237593 and A245092.
(End)

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 7342 terms from Robert Price)
Omar E. Pol, Illustration of a(11) = 1736, Perspective view of the stepped pyramid with 11 levels which contains 1736 unit cubes.

Cf. A000203, A024916, A046092, A008586, A175254, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239050, A239660, A239931, A239932, A239933, A239934, A243980, A245092, A262626, A340793.

[4*(&+[(n-k+1)*DivisorSigma(1,k): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Apr 07 2019

a[n_] := 4 Sum[(n - k + 1) DivisorSigma[1, k], {k, n}]; Array[a, 40] (* Robert G. Wilson v, Aug 06 2018 *)
Nest[Accumulate,4*DivisorSigma[1,Range[50]],2] (* Harvey P. Dale, Sep 07 2022 *)

a(n) = 4*sum(k=1, n, sigma(k)*(n-k+1)); \\ Michel Marcus, Aug 07 2018

from math import isqrt
def A244050(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1))<<1)//3 # Chai Wah Wu, Oct 22 2023

[4*sum(sigma(k)*(n-k+1) for k in (1..n)) for n in (1..40)] # G. C. Greubel, Apr 07 2019

a(n) = 4*A175254(n).

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

Original entry on oeis.org

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).

A239050 a(n) = 4*sigma(n).

Original entry on oeis.org

4, 12, 16, 28, 24, 48, 32, 60, 52, 72, 48, 112, 56, 96, 96, 124, 72, 156, 80, 168, 128, 144, 96, 240, 124, 168, 160, 224, 120, 288, 128, 252, 192, 216, 192, 364, 152, 240, 224, 360, 168, 384, 176, 336, 312, 288, 192, 496, 228, 372, 288, 392, 216, 480, 288, 480, 320, 360, 240, 672, 248, 384, 416, 508

Offset: 1

Omar E. Pol, Mar 09 2014

4 times the sum of divisors of n.

a(n) is also the total number of horizontal cells in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every three-dimensional quadrant arises after the 90-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a square formed by four cells (see links and examples). - Omar E. Pol, Jul 04 2016

			For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28.
For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24.
.
Illustration of initial terms:                                    _ _ _ _ _ _
.                                           _ _ _ _ _ _          |_|_|_|_|_|_|
.                           _ _ _ _       _|_|_|_|_|_|_|_     _ _|           |_ _
.             _ _ _ _     _|_|_|_|_|_    |_|_|       |_|_|   |_|               |_|
.     _ _    |_|_|_|_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|   |_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|_ _|_|   |_|       |_|   |_|           |_|   |_|               |_|
.            |_|_|_|_|   |_|_ _ _ _|_|   |_|_         _|_|   |_|               |_|
.                          |_|_|_|_|     |_|_|_ _ _ _|_|_|   |_|_             _|_|
.                                          |_|_|_|_|_|_|         |_ _ _ _ _ _|
.                                                                |_|_|_|_|_|_|
.
n:     1          2             3                4                     5
S(n):  1          3             4                7                     6
a(n):  4         12            16               28                    24
.
For n = 1..5, the figure n represents the reflection in the four quadrants of the symmetric representation of S(n) = sigma(n) = A000203(n). For more information see A237270 and A237593.
The diagram also represents the top view of the first four terraces of the stepped pyramid described in Comments section. - _Omar E. Pol_, Jul 04 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Folding the first eight rows of triangle
Index entries for sequences related to sigma(n)

Alternating row sums of A239662.
Partial sums give A243980.
k times sigma(n), k=1..6: A000203, A074400, A272027, this sequence, A274535, A274536.
k times sigma(n), k = 1..10: A000203, A074400, A272027, this sequence, A274535, A274536, A319527, A319528, A325299, A326122.
Cf. A008438, A017113, A062731, A112610, A144613, A193553, A196020, A235791, A236104, A237270, A237593, A239052, A239053, A239660, A239662, A244050, A262626.

[4*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jul 30 2019

with(numtheory): seq(4*sigma(n), n=1..64); # Omar E. Pol, Jul 04 2016

Array[4 DivisorSigma[1, #] &, 64] (* Michael De Vlieger, Nov 16 2017 *)

a(n) = 4 * sigma(n); \\ Omar E. Pol, Jul 04 2016

a(n) = 4*A000203(n) = 2*A074400(n).
a(n) = A000203(n) + A272027(n). - Omar E. Pol, Jul 04 2016
Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016
Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - Omar E. Pol, Oct 02 2018
The conjecture above is correct. Write n = 3^e*m, gcd(3, m) = 1, then sigma(3*n) = sigma(3^(e+1))*sigma(m) = ((3^(e+2) - 1)/2)*sigma(m) = ((3^(e+2) - 1)/(3^(e+1) - 1))*sigma(3^e*m), and (3^(e+2) - 1)/(3^(e+1) - 1) = 4 if and only if e = 0. - Jianing Song, Feb 03 2019

A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

Original entry on oeis.org

8, 24, 48, 80, 112, 160, 200, 264, 328, 408, 464, 560, 624, 728, 832, 960, 1040, 1184, 1272, 1432, 1576, 1728, 1832, 2024, 2160, 2336, 2512, 2736

Offset: 1

Omar E. Pol, Jun 26 2014

Partial sums of A244371.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).

			Illustration of the structure after 16 stages (Contains 960 toothpicks):
.
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244371, A244970, A244971, A245092.

a(n) = 4*A244362(n) = 8*A244360(n).

a(8) corrected and more terms from Omar E. Pol, Oct 18 2014

A327329 Twice the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

2, 8, 16, 30, 42, 66, 82, 112, 138, 174, 198, 254, 282, 330, 378, 440, 476, 554, 594, 678, 742, 814, 862, 982, 1044, 1128, 1208, 1320, 1380, 1524, 1588, 1714, 1810, 1918, 2014, 2196, 2272, 2392, 2504, 2684, 2768, 2960, 3048, 3216, 3372, 3516, 3612, 3860, 3974, 4160, 4304, 4500, 4608, 4848, 4992

Offset: 1

Omar E. Pol, Sep 25 2019

nonn

a(n) has a symmetric representation. Using two opposite quadrants, where in each quadrant there is the Dyck path related to partitions described in the n-th row of triangle A237593, a(n) is the total area (or the total number of cells) of the structure (see the example).

a(n) is also the total area of the horizontal faces in the stepped pyramid with n levels described in A245092 (that is the total area of the terraces plus the area of the base). - Omar E. Pol, Dec 15 2021

			Illustration of a(8) = 112 using a symmetric structure constructed with the Dyck path related to partitions described in the 8th row of triangle A237593.
                           _ _ _ _ _
                          |         |
                          |         |_
                          |           |_ _
                          |               |
                          |     56        |
                          |               |
                          |               |
           _ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _|
          |               |
          |               |
          |               |
          |       56      |
          |_ _            |
              |_          |
                |         |
                |_ _ _ _ _|

Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Partial sums of A074400.

Cf. A001105, A006218, A013661, A024916, A067436, A222548, A236104, A237591, A237593, A243980, A245092, A262626.

Accumulate[2*DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Sep 25 2021 *)

a(n) = 2*sum(k=1, n, sigma(k)); \\ Michel Marcus, Dec 20 2021

from sympy import divisor_sigma
from itertools import accumulate
def f(, n): return  + 2*divisor_sigma(n, 1)
def aupton(terms): return list(accumulate(range(terms+1), f))[1:]
print(aupton(55)) # Michael S. Branicky, Dec 16 2021

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)) # Chai Wah Wu, Oct 22 2023

a(n) = 2*A024916(n).
a(n) = A243980(n)/2.
a(n) = A006218(n) + A222548(n).
a(n) = A001105(n) - A067436(n).
lim_{n->infinity} a(n)/(n^2) = Pi^2/6 = zeta(2) (cf. A013661). - Omar E. Pol, Dec 16 2021

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.

A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

Original entry on oeis.org

1, 2, 6, 7, 11, 12, 16, 17, 25, 29, 33, 34, 38, 42, 50, 51, 55, 56, 60, 61, 73, 77, 81, 82, 90, 94, 106, 107, 111, 112, 116, 117, 129, 133, 141, 142, 146, 150, 162, 163, 167, 168, 172, 176, 184, 188, 192, 193, 201, 209, 221, 225, 229, 230, 242, 243, 255, 259, 263, 264

Offset: 1

Omar E. Pol, Jul 08 2014

nonn

Partial sums of A244971.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A244050. - Omar E. Pol, Apr 20 2016

			Illustration of the structure after 15 stages (contains 50 regions):
.
.                   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.               _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _
.             _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_
.           _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_
.          |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  |
.     _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _
.    | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | |
.    | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | |
.    | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | |
.    | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | |
.    | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | |
.    | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | |
.    | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | |
.    | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | |
.    | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | |
.    | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | |
.    | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | |
.    | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | |
.    | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | |
.    |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_|
.          | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |
.          |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _|
.            |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|
.              |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|
.                  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram is also the top view of the stepped pyramid with 15 levels described in A244050. - _Omar E. Pol_, Apr 20 2016

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237590, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244971, A245092.

A294629 Partial sums of A294628.

Original entry on oeis.org

4, 16, 28, 56, 68, 120, 132, 192, 228, 296, 308, 440, 452, 536, 612, 736, 748, 920, 932, 1112, 1204, 1320, 1332, 1624, 1676, 1808, 1916, 2144, 2156, 2496, 2508, 2760, 2884, 3048, 3156, 3600, 3612, 3792, 3932, 4336, 4348, 4784, 4796, 5120, 5388, 5600, 5612, 6224, 6292, 6640, 6812, 7184, 7196, 7728, 7868, 8384

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the stepped pyramid described in A294630.

Number of terms less than 10^k, k=1,2,3,...: 1, 5, 19, 61, 195, 623, 1967, 6225, ... - Muniru A Asiru, Mar 04 2018

			Illustration of initial terms (n = 1..6):
.                                                  _ _ _ _ _ _
.                                _ _ _ _         _|     |     |_
.                _ _ _ _       _|   |   |_      |       |       |
.      _ _      |   |   |     |    _|_    |     |      _|_      |
.     |_|_|     |_ _|_ _|     |_ _|   |_ _|     |_ _ _|   |_ _ _|
.     |_|_|     |   |   |     |   |_ _|   |     |     |_ _|     |
.               |_ _|_ _|     |_    |    _|     |       |       |
.       4                       |_ _|_ _|       |_      |      _|
.                  16                             |_ _ _|_ _ _|
.                                  28
.                                                      56
.
.                                        _ _ _ _ _ _ _ _
.             _ _ _ _ _ _              _|       |       |_
.            |     |     |           _|         |         |_
.         _ _|     |     |_ _       |           |           |
.        |      _ _|_ _      |      |          _|_          |
.        |     |       |     |      |        _|   |_        |
.        |_ _ _|       |_ _ _|      |_ _ _ _|       |_ _ _ _|
.        |     |       |     |      |       |_     _|       |
.        |     |_ _ _ _|     |      |         |_ _|         |
.        |_ _      |      _ _|      |           |           |
.            |     |     |          |_          |          _|
.            |_ _ _|_ _ _|            |_        |        _|
.                                       |_ _ _ _|_ _ _ _|
.                 68
.                                              120
.
Note that for n >= 2 the structure has a hole (or hollow) in the center.
a(n) is the number of ON cells in the n-th diagram.

Iain Fox, Table of n, a(n) for n = 1..10000

For other related diagrams see A294630 (partial sums), A294016 and A237593.

Cf. A000203, A004125, A016742, A024916, A067436, A239050, A243980, A294015, A294017, A294628.

List([1..1000],n->Sum([1..n],k->8*(Sigma(k)-k+(1/2)))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sum(8*(sigma(k)-k+(1/2)),k=1..n),n=1..1000); # Muniru A Asiru, Mar 04 2018

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@Array[f, 56] (* Robert G. Wilson v, Dec 12 2017 *)

a(n) = 4*(sum(k=1, n, n\k*k) - sum(k=2, n, n%k)) \\ Iain Fox, Dec 10 2017

first(n) = my(res = vector(n)); res[1] = 4; for(x=2, n, res[x] = res[x-1] + 8*(sigma(x) - x + (1/2))); res; \\ Iain Fox, Dec 10 2017

from math import isqrt
def A294629(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-n**2<<2 # Chai Wah Wu, Oct 22 2023

a(n) = 4*A294016(n).
a(n) = A016742(n) - 8*A004125(n).
a(n) = A016742(n) - 4*A067436(n).
a(n) = A243980(n) - 4*A004125(n).
a(n) = A243980(n) - 2*A067436(n).

A294630 Partial sums of A294629.

Original entry on oeis.org

4, 20, 48, 104, 172, 292, 424, 616, 844, 1140, 1448, 1888, 2340, 2876, 3488, 4224, 4972, 5892, 6824, 7936, 9140, 10460, 11792, 13416, 15092, 16900, 18816, 20960, 23116, 25612, 28120, 30880, 33764, 36812, 39968, 43568, 47180, 50972, 54904, 59240, 63588, 68372, 73168, 78288, 83676, 89276, 94888, 101112

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

a(n) is also the volume of a stepped pyramid with n levels which is another version of the stepped pyramid described in A244050. Both pyramids have the same top view and the same front view, that is to say externally both pyramids are equal, but this pyramid with n levels contains a central chamber whose volume is 4*A072481(n). For more information about the central chamber see the diagrams in A294629.

a(n) is the number of unit cubes of the pyramid with n levels.

			Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid. For more information about the hidden pattern see A237593 and A245092.

Muniru A Asiru, Table of n, a(n) for n = 1..2000

For other related diagrams see A294016 and A294629.

Cf. A000203, A002492, A004125, A072481, A237593, A239050, A243980, A244050, A245092, A294015, A294017, A294628.

List([1..50],n->Sum([1..n],m->Sum([1..m],k->8*(Sigma(k)-k+(1/2))))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sum(sum(8*(sigma(j)-j+(1/2)),j=1..k),k=1..n),n=1..50); # Muniru A Asiru, Mar 04 2018

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* Robert G. Wilson v, Dec 12 2017 *)

from math import isqrt
def A294630(n): return ((((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1, s+1))<<2)-(n*(n+1)*((n<<1)+1)<<1))//3 # Chai Wah Wu, Nov 01 2023

a(n) = 4*A294017(n).
a(n) = A002492(n) - 8*A072481(n).
a(n) = A244050(n) - 4*A072481(n).

cp's OEIS Frontend

A244050 Partial sums of A243980.

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

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A239050 a(n) = 4*sigma(n).

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A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

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A327329 Twice 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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A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

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A294629 Partial sums of A294628.

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