A274536 -id:A274536 - cp's OEIS frontend

A074400 Sum of the even divisors of 2n.

2, 6, 8, 14, 12, 24, 16, 30, 26, 36, 24, 56, 28, 48, 48, 62, 36, 78, 40, 84, 64, 72, 48, 120, 62, 84, 80, 112, 60, 144, 64, 126, 96, 108, 96, 182, 76, 120, 112, 180, 84, 192, 88, 168, 156, 144, 96, 248, 114, 186, 144, 196, 108, 240, 144, 240, 160, 180, 120, 336, 124, 192

Offset: 1

Joseph L. Pe, Nov 25 2002

Also alternating row sums of A236106. - Omar E. Pol, Jan 23 2014

Could also be called the twice sigma function, see first formula. - Omar E. Pol, Feb 05 2014

			The even divisors of 12 are 12, 6, 4, 2, which sum to 24, so a(6) = 24.

k times sigma(n), k=1..6: A000203, this sequence, A272027, A239050, A274535, A274536.

Cf. A146076, which includes the zeros for odd n.

with(numtheory): seq(2*sigma(n),n=1..65);

f[n_] := Plus @@ Select[ Divisors[ 2n], EvenQ]; Array[f, 62] (* Robert G. Wilson v, Apr 09 2011 *)

a(n) = 2 * sigma(n); \\ Joerg Arndt, Apr 14 2013

a(n) = sumdiv(2*n, d, !(d%2) * d); \\ Michel Marcus, Jan 23 2014

a(n) = 2*sigma(n) = 2*A000203(n).

Dirichlet g.f.: 2*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 06 2016

More terms from Emeric Deutsch, May 24 2004

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

A272027 a(n) = 3*sigma(n).

Original entry on oeis.org

3, 9, 12, 21, 18, 36, 24, 45, 39, 54, 36, 84, 42, 72, 72, 93, 54, 117, 60, 126, 96, 108, 72, 180, 93, 126, 120, 168, 90, 216, 96, 189, 144, 162, 144, 273, 114, 180, 168, 270, 126, 288, 132, 252, 234, 216, 144, 372, 171, 279, 216, 294, 162, 360, 216, 360, 240, 270, 180, 504, 186, 288, 312, 381

Offset: 1

Omar E. Pol, Apr 18 2016

3 times the sum of the divisors of n.
From Omar E. Pol, Jul 04 2016: (Start)
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 120-degree three-dimensional sector arises after the 120-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a hexagon formed by three rhombuses (see Links section).
More generally, if k >= 3 then k*sigma(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid where the structure of every 360/k three-dimensional sector arises after the 360/k-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. If k >= 5 the top of the pyramid is a k-pointed star formed by k rhombuses. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle before the 120-degree-zig-zag folding (rows: 1..28)
Index entries for sequences related to sigma(n)

Alternating row sums of triangle A272026.
k times sigma(n), k = 1..10: A000203, A074400, this sequence, A239050, A274535, A274536, A319527, A319528, A325299, A326122.
Cf. A062731, A196020, A236104, A237270, A237593.

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

with(numtheory): seq(3*sigma(n), n=1..64);

Table[3 DivisorSigma[1, n], {n, 64}] (* Michael De Vlieger, Apr 19 2016 *)

PARI
```
a(n) = 3 * sigma(n);
```

a(n) = 3*A000203(n) = A000203(n) + A074400(n) = A239050(n) - A000203(n).
Dirichlet g.f.: 3*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016
a(n) = A274536(n)/2. - Antti Karttunen, Nov 16 2017
From Omar E. Pol, Oct 02 2018: (Start)
Conjecture 1: a(n) = sigma(2*n) = A062731(n) iff n is odd.
And more generally:
Conjecture 2: If p is prime then (p + 1)*sigma(n) = sigma(p*n) iff n is not a multiple of p. (End)
The above claims easily follow from the fact that sigma is multiplicative function, thus if p does not divide n, then sigma(p*n) = sigma(p)*sigma(n). - Antti Karttunen, Nov 21 2019

A274535 a(n) = 5*sigma(n).

Original entry on oeis.org

5, 15, 20, 35, 30, 60, 40, 75, 65, 90, 60, 140, 70, 120, 120, 155, 90, 195, 100, 210, 160, 180, 120, 300, 155, 210, 200, 280, 150, 360, 160, 315, 240, 270, 240, 455, 190, 300, 280, 450, 210, 480, 220, 420, 390, 360, 240, 620, 285, 465, 360, 490, 270, 600, 360, 600, 400, 450, 300, 840, 310, 480, 520, 635

Offset: 1

Omar E. Pol, Jun 29 2016

5 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 72-degree-three-dimensional sector arises after the 72-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a five-pointed star formed by five rhombuses (see Links section).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle before the 72-degree-zig-zag folding (rows: 1..28)
Index entries for sequences related to sigma(n)

k times sigma(n), k=1..6: A000203, A074400, A272027, A239050, this sequence, A274536.

Cf. A237593.

with(numtheory): seq(5*sigma(n), n=1..64);

Table[5 DivisorSigma[1, n], {n, 64}] (* Michael De Vlieger, Jul 04 2016 *)

PARI
```
a(n) = 5 * sigma(n);
```

a(n) = 5*A000203(n) = A000203(n) + A239050(n) = A074400(n) + A272027(n).

Dirichlet g.f.: 5*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016

A319527 a(n) = 7 * sigma(n).

Original entry on oeis.org

7, 21, 28, 49, 42, 84, 56, 105, 91, 126, 84, 196, 98, 168, 168, 217, 126, 273, 140, 294, 224, 252, 168, 420, 217, 294, 280, 392, 210, 504, 224, 441, 336, 378, 336, 637, 266, 420, 392, 630, 294, 672, 308, 588, 546, 504, 336, 868, 399, 651, 504, 686, 378, 840, 504, 840, 560, 630, 420, 1176, 434, 672, 728, 889

Offset: 1

Omar E. Pol, Sep 22 2018

7 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 (360/7)-degree-three-dimensional sector arises after the (360/7)-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a seven-pointed star formed by seven rhombuses (see Links section).

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

Cf. A216606, A237593.

List([1..70],n->7*Sigma(n)); # Muniru A Asiru, Sep 28 2018

with(numtheory): seq(7*sigma(n), n=1..64);

7*DivisorSigma[1,Range[70]] (* Harvey P. Dale, Mar 14 2020 *)

PARI
```
a(n) = 7 * sigma(n);
```

a(n) = 7*A000203(n).
a(n) = A000203(n) + A274536(n).
Dirichlet g.f.: 7*zeta(s-1)*zeta(s). - (After Ilya Gutkovskiy)

A319528 a(n) = 8 * sigma(n).

Original entry on oeis.org

8, 24, 32, 56, 48, 96, 64, 120, 104, 144, 96, 224, 112, 192, 192, 248, 144, 312, 160, 336, 256, 288, 192, 480, 248, 336, 320, 448, 240, 576, 256, 504, 384, 432, 384, 728, 304, 480, 448, 720, 336, 768, 352, 672, 624, 576, 384, 992, 456, 744, 576, 784, 432, 960, 576, 960, 640, 720, 480, 1344, 496, 768, 832

Offset: 1

Omar E. Pol, Sep 22 2018

8 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 45-degree three-dimensional sector arises after the 45-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is an eight-pointed star formed by eight rhombuses (see Links section).

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle A237593 before the 45-degree-zig-zag folding (rows: 1..28)
Index entries for sequences related to sigma(n)

k times sigma(n), k=1..7: A000203, A074400, A272027, A239050, A274535, A274536, A319527.

Cf. A008589, A047304, A237593, A283078.

List([1..70],n->8*Sigma(n)); # Muniru A Asiru, Sep 28 2018

with(numtheory): seq(8*sigma(n), n=1..64);

8*DivisorSigma[1,Range[70]] (* Harvey P. Dale, Dec 24 2018 *)

PARI
```
a(n) = 8 * sigma(n);
```

a(n) = 8*A000203(n) = 4*A074400(n) = 2*A239050(n).
a(n) = A000203(n) + A319527(n).
Dirichlet g.f.: 8*zeta(s-1)*zeta(s). (After Ilya Gutkovskiy)
Conjecture: a(n) = sigma(7*n) = A283078(n) iff n is not a multiple of 7.
Conjecture is true, since sigma is multiplicative, so if (7,n) = 1 then sigma(7*n) = sigma(7)*sigma(n) = 8*sigma(n). - Charlie Neder, Oct 02 2018

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

Original entry on oeis.org

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 *)

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)

A383403 Partial sums of the sum of the divisors of the numbers of the form 6*k + 3, k >= 0.

Original entry on oeis.org

4, 17, 41, 73, 113, 161, 217, 295, 367, 447, 551, 647, 771, 892, 1012, 1140, 1296, 1488, 1640, 1822, 1990, 2166, 2406, 2598, 2826, 3060, 3276, 3564, 3824, 4064, 4312, 4632, 4968, 5240, 5552, 5840, 6136, 6539, 6923, 7243, 7607, 7943, 8375, 8765, 9125, 9573, 9989, 10469, 10861

Offset: 0

Omar E. Pol, Apr 27 2025

Partial sums of the sum of the divisors of A016945.
See the illustration of a(3) and a(10) as the total area (or total number of cells) in the diagram of the symmetric representation of sigma in the Links section.
Also consider a spiral similar to the spiral described in A239660 but with six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the total number of diamonds (or the total area) in the third wedge after n + 1 turns. The spiral can be visualized from the top view of the stepped pyramid described in A274536. The graph is named W3 in the Plot 6 of the Links section.

			For n = 3 the first four terms of the numbers of the form 6*k + 3, k >= 0, are [3, 9, 15, 21]. The divisors of them are [1, 3], [1, 3, 9], [1, 3, 5, 15], [1, 3, 7, 21]. The sum of the divisors of them are [4, 13, 24, 32] respectively, and the sum of all divisors of them are 4 + 13 + 24 + 32 = 73, so a(3) = 73.

Omar E. Pol, Illustration of a(3) = 73
Omar E. Pol, Illustration of a(10) = 551
Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Sequences of the same family are A363161, A365442, this sequence, A365444, A383405, A365446.

Cf. A000203, A016945, A237593, A239660, A274536.

Accumulate@ Array[DivisorSigma[1, 6 # + 3] &, 55, 0]

PARI
```
a(n) = sum(k=0, n, sigma(6*k+3));
```

a(n) = Sum_{k=0..n} sigma(6*k+3).

a(n) = (11*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Apr 28 2025

cp's OEIS Frontend

A074400 Sum of the even divisors of 2n.

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

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Magma

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A272027 a(n) = 3*sigma(n).

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A274535 a(n) = 5*sigma(n).

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A319527 a(n) = 7 * sigma(n).

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A319528 a(n) = 8 * sigma(n).

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