A239050 -id:A239050 - cp's OEIS frontend

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 4, 6, 3, 7, 8, 9, 4, 6, 14, 12, 12, 5, 12, 12, 21, 16, 15, 6, 8, 24, 18, 28, 20, 18, 7, 15, 16, 36, 24, 35, 24, 21, 8, 13, 30, 24, 48, 30, 42, 28, 24, 9, 18, 26, 45, 32, 60, 36, 49, 32, 27, 10, 12, 36, 39, 60, 40, 72, 42, 56, 36, 30, 11, 28, 24, 54, 52, 75, 48, 84, 48, 63, 40, 33, 12

Offset: 1

Omar E. Pol, Sep 22 2018

			The corner of the square array begins:
         A000203 A074400 A272027 A239050 A274535 A274536 A319527 A319528
A000027:       1,      2,      3,      4,      5,      6,      7,      8, ...
A008585:       3,      6,      9,     12,     15,     18,     21,     24, ...
A008586:       4,      8,     12,     16,     20,     24,     28,     32, ...
A008589:       7,     14,     21,     28,     35,     42,     49,     56, ...
A008588:       6,     12,     18,     24,     30,     36,     42,     48, ...
A008594:      12,     24,     36,     48,     60,     72,     84,     96, ...
A008590:       8,     16,     24,     32,     40,     48,     56,     64, ...
A008597:      15,     30,     45,     60,     75,     90,    105,    120, ...
A008595:      13,     26,     39,     52,     65,     78,     91,    104, ...
A008600:      18,     36,     54,     72,     90,    108,    126,    144, ...
...

Index entries for sequences related to sigma(n)

Another version of A274824.
Antidiagonal sums give A175254.
Main diagonal gives A064987.
Row n lists the multiples of A000203(n).
Row 1 is A000027.
Initial zeros should be omitted in the following sequences related to the rows of the array:
Row 2-5: A008585, A008586, A008589, A008588.
Rows 6 and 11: A008594.
Rows 7-9: A008590, A008597, A008595.
Rows 10 and 17: A008600.
Rows 12-13: A135628, A008596.
Rows 14, 15 and 23: A008606.
Rows 16 and 25: A135631.
(Note that in the OEIS there are many other sequences that are also rows of this square array.)
Cf. A000203, A237593, A319526.

T:=Flat(List([1..12],n->List([1..n],k->k*Sigma(n-k+1))));; Print(T); # Muniru A Asiru, Jan 01 2019

with(numtheory): T:=(n,k)->k*sigma(n-k+1): seq(seq(T(n,k),k=1..n),n=1..12); # Muniru A Asiru, Jan 01 2019

Table[k DivisorSigma[1, #] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

Original entry on oeis.org

1, 0, 3, 1, 0, 4, 1, 3, 0, 7, 2, 3, 4, 0, 6, 2, 6, 4, 7, 0, 12, 4, 6, 8, 7, 6, 0, 8, 4, 12, 8, 14, 6, 12, 0, 15, 7, 12, 16, 14, 12, 12, 8, 0, 13, 8, 21, 16, 28, 12, 24, 8, 15, 0, 18, 12, 24, 28, 28, 24, 24, 16, 15, 13, 0, 12, 14, 36, 32, 49, 24, 48, 16, 30, 13, 18, 0, 28

Offset: 1

Omar E. Pol, Jan 15 2021

T(n,k) is the total number of cubic cells added at n-th stage to the right prisms whose bases are the parts of the symmetric representation of sigma(k) in the polycube described in A221529.

Partial sums of column k gives the column k of A221529.

			Triangle begins:
   1;
   0,  3;
   1,  0,  4;
   1,  3,  0,  7;
   2,  3,  4,  0,  6;
   2,  6,  4,  7,  0, 12;
   4,  6,  8,  7,  6,  0,  8;
   4, 12,  8, 14,  6, 12,  0, 15;
   7, 12, 16, 14, 12, 12,  8,  0, 13;
   8, 21, 16, 28, 12, 24,  8, 15,  0, 18;
  12, 24, 28, 28, 24, 24, 16, 15, 13,  0, 12;
  14, 36, 32, 49, 24, 48, 16, 30, 13, 18,  0, 28;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000203         T(6,k)
--------------------------
1      1   *   2  =    2
2      3   *   2   =   6
3      4   *   1   =   4
4      7   *   1   =   7
5      6   *   0   =   0
6     12   *   1   =  12
.           A002865
--------------------------
The sum of row 6 is 2 + 6 + 4 + 7 + 0 + 12 = 31, equaling A138879(6).

Row sums give A138879.
Column 1 gives A002865.
Diagonals 1, 3 and 4 give A000203.
Diagonal 2 gives A000004.
Diagonals 5 and 6 give A074400.
Diagonals 7 and 8 give A239050.
Diagonal 9 gives A319527.
Diagonal 10 gives A319528.
Cf. A221529 (partial column sums).
Cf. A340426 (mirror).
Cf. A135010, A221530, A245095, A245099, A339278.

A340583[n_, k_] := (PartitionsP[n - k] - PartitionsP[(n - k) - 1])*
   DivisorSigma[1, k];
Table[A340583[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Robert P. P. McKone, Jan 25 2021 *)

A294628 a(n) = 8*(sigma(n) - n + (1/2)).

Original entry on oeis.org

4, 12, 12, 28, 12, 52, 12, 60, 36, 68, 12, 132, 12, 84, 76, 124, 12, 172, 12, 180, 92, 116, 12, 292, 52, 132, 108, 228, 12, 340, 12, 252, 124, 164, 108, 444, 12, 180, 140, 404, 12, 436, 12, 324, 268, 212, 12, 612, 68, 348, 172, 372, 12, 532, 140, 516, 188, 260, 12, 868, 12, 276

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

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

Partial sums give A294629.

Cf. A001065, A017113, A235796, A237593, A239050, A294015, A294016, A294017, A294630.

List([1..10^5],n->8*(Sigma(n)-n+(1/2))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sigma(8*n-1)/8, n=1..10^3); # Muniru A Asiru, Mar 04 2018

a[n_] := 8 (DivisorSigma[1, n] - n) + 4; Array[a, 62] (* Robert G. Wilson v, Dec 12 2017 *)

a(n) = 4*A294015(n).
a(n) = 8*(A001065(n) + (1/2)).
a(n) = 8*(A000203(n) - n + (1/2)).
a(n) = A239050(n) - 4*A235796(n).
a(n) = A017113(n-1) - 8*A235796(n).

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).

A325299 a(n) = 9 * sigma(n).

Original entry on oeis.org

9, 27, 36, 63, 54, 108, 72, 135, 117, 162, 108, 252, 126, 216, 216, 279, 162, 351, 180, 378, 288, 324, 216, 540, 279, 378, 360, 504, 270, 648, 288, 567, 432, 486, 432, 819, 342, 540, 504, 810, 378, 864, 396, 756, 702, 648, 432, 1116, 513, 837, 648, 882, 486, 1080, 648, 1080, 720, 810, 540, 1512

Offset: 1

Omar E. Pol, Jun 26 2019

9 times the sum of the divisors of n.

a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) in which the structure of every 40-degree-three-dimensional sector arises after the 40-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a nine-pointed star formed by nine rhombuses (see Links section).

k times sigma(n), k=1..8: A000203, A074400, A272027, A239050, A274535, A274536, A319527, A319528.

Cf. A008591, A237593.

List([1..70],n->9*Sigma(n)); # After Muniru A Asiru

with(numtheory): seq(9*sigma(n), n=1..64);

9*DivisorSigma[1,Range[70]] (* After Harvey P. Dale *)

PARI
```
a(n) = 9 * sigma(n);
```

a(n) = 9*A000203(n) = 3*A272027(n).
a(n) = A000203(n) + A319528(n) = A074400(n) + A319527(n).
Dirichlet g.f.: 9*zeta(s-1)*zeta(s). - (After Ilya Gutkovskiy)

A326122 a(n) = 10 * sigma(n).

Original entry on oeis.org

10, 30, 40, 70, 60, 120, 80, 150, 130, 180, 120, 280, 140, 240, 240, 310, 180, 390, 200, 420, 320, 360, 240, 600, 310, 420, 400, 560, 300, 720, 320, 630, 480, 540, 480, 910, 380, 600, 560, 900, 420, 960, 440, 840, 780, 720, 480, 1240, 570, 930, 720, 980, 540, 1200, 720, 1200, 800, 900, 600, 1680, 620, 960

Offset: 1

Omar E. Pol, Jul 13 2019

10 times the sum of the divisors of n.

a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every 36-degree-three-dimensional sector arises after the 36-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a 10-pointed star formed by 10 rhombuses (see Links section).

k times sigma(n), k=1..9: A000203, A074400, A272027, A239050, A274535, A274536, A319527, A319528, A325299.

Cf. A008592, A033583, A124080, A124252, A142342, A153780, A237593.

List([1..70],n->10*Sigma(n)); # After Muniru A Asiru

[10*DivisorSigma(1, n): n in [1..70]]; // Vincenzo Librandi, Jul 26 2019

with(numtheory): seq(10*sigma(n), n=1..64);

10*DivisorSigma[1,Range[70]] (* After Harvey P. Dale *)

PARI
```
a(n) = 10 * sigma(n);
```

a(n) = 10*A000203(n) = 5*A074400(n) = 2*A274535(n).
a(n) = A000203(n) + A325299(n) = A074400(n) + A319528(n).
Dirichlet g.f.: 10*zeta(s-1)*zeta(s). - (After Ilya Gutkovskiy)

A380231 Alternating row sums of triangle A237591.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 3, 4, 5, 4, 3, 6, 5, 4, 7, 8, 7, 8, 7, 10, 9, 8, 7, 10, 11, 10, 9, 12, 11, 14, 13, 14, 13, 12, 15, 16, 15, 14, 13, 16, 15, 18, 17, 16, 19, 18, 17, 20, 21, 22, 21, 20, 19, 22, 21, 24, 23, 22, 21, 24, 23, 22, 25, 26, 25, 28, 27, 26, 25, 28, 27, 32, 31, 30, 29, 28, 31, 30, 29

Offset: 1

Omar E. Pol, Jan 17 2025

nonn

Consider the symmetric Dyck path in the first quadrant of the square grid described in the n-th row of A237593. Let C = (A240542(n), A240542(n)) be the middle point of the Dyck path.
a(n) is also the coordinate on the x axis of the point (a(n),n) and also the coordinate on the y axis of the point (n,a(n)) such that the middle point of the line segment [(a(n),n),(n,a(n))] coincides with the middle point C of the symmetric Dyck path.
The three line segments [(a(n),n),C], [(n,a(n)),C] and [(n,n),C] have the same length.
For n > 2 the points (n,n), C and (a(n),n) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (n,n), C and (n,a(n)) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (a(n),n), (n,n) and (n,a(n)) are the vertices of a virtual isosceles right triangle.

			For n = 14 the 14th row of A237591 is [8, 3, 1, 2] hence the alternating row sum is 8 - 3 + 1 - 2 = 4, so a(14) = 4.
On the other hand the 14th row of A237593 is the 14th row of A237591 together with the 14 th row of A237591 in reverse order as follows: [8, 3, 1, 2, 2, 1, 3, 8].
Then with the terms of the 14th row of A237593 we can draw a Dyck path in the first quadrant of the square grid as shown below:
.
         (y axis)
          .
          .
          .    (4,14)              (14,14)
          ._ _ _ . _ _ _ _            .
          .               |
          .               |
          .               |_
          .                 |
          .                 |_ _
          .                C    |_ _ _
          .                           |
          .                           |
          .                           |
          .                           |
          .                           . (14,4)
          .                           |
          .                           |
          . . . . . . . . . . . . . . | . . . (x axis)
        (0,0)
.
In the example the point C is the point (9,9).
The three line segments [(4,14),(9,9)], [(14,4),(9,9)] and [(14,14),(9,9)] have the same length.
The points (14,14), (9,9) and (4,14) are the vertices of a virtual isosceles right triangle.
The points (14,14), (9,9) and (14,4) are the vertices of a virtual isosceles right triangle.
The points (4,14), (14,14) and (14,4) are the vertices of a virtual isosceles right triangle.

Other alternating row sums (ARS) related to the Dyck paths of A237593 and the stepped pyramid described in A245092 are as follows:
ARS of A237593 give A000004.
ARS of A196020 give A000203.
ARS of A252117 give A000203.
ARS of A271343 give A000593.
ARS of A231347 give A001065.
ARS of A236112 give A004125.
ARS of A236104 give A024916.
ARS of A249120 give A024916.
ARS of A271344 give A033879.
ARS of A231345 give A033880.
ARS of A239313 give A048050.
ARS of A237048 give A067742.
ARS of A236106 give A074400.
ARS of A235794 give A120444.
ARS of A266537 give A146076.
ARS of A236540 give A153485.
ARS of A262612 give A175254.
ARS of A353690 give A175254.
ARS of A239446 give A235796.
ARS of A239662 give A239050.
ARS of A235791 give A240542.
ARS of A272026 give A272027.
ARS of A211343 give A336305.
Cf. A221529, A237270, A237271, A237591, A262626, A322141.

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
a(n) = my(orow = concat(row235791(n), 0)); vecsum(vector(#orow-1, i, (-1)^(i+1)*(orow[i] - orow[i+1]))); \\ Michel Marcus, Apr 13 2025

a(n) = 2*A240542(n) - n.
a(n) = n - 2*A322141(n).
a(n) = A240542(n) - A322141(n).

cp's OEIS Frontend

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

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A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

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A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

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A294628 a(n) = 8*(sigma(n) - n + (1/2)).

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A294629 Partial sums of A294628.

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A294630 Partial sums of A294629.

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A325299 a(n) = 9 * sigma(n).

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