A319528 -id:A319528 - cp's OEIS frontend

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

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

A274536 a(n) = 6 * sigma(n).

Original entry on oeis.org

6, 18, 24, 42, 36, 72, 48, 90, 78, 108, 72, 168, 84, 144, 144, 186, 108, 234, 120, 252, 192, 216, 144, 360, 186, 252, 240, 336, 180, 432, 192, 378, 288, 324, 288, 546, 228, 360, 336, 540, 252, 576, 264, 504, 468, 432, 288, 744, 342, 558, 432, 588, 324, 720, 432, 720, 480, 540, 360, 1008, 372, 576, 624, 762

Offset: 1

Omar E. Pol, Jun 29 2016

6 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 60-degree-three-dimensional sector arises after the 60-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a six-pointed star formed by six rhombuses (see Links section).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle before the 60-degree-zig-zag folding (rows: 1..28)
Index entries for sequences related to sigma(n)

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

Cf. A237593, A283118.

with(numtheory): seq(6*sigma(n), n=1..64);

6DivisorSigma[1, Range[50]] (* Alonso del Arte, Jul 04 2016 *)

PARI
```
a(n) = 6 * sigma(n);
```

a(n) = 6*A000203(n) = 3*A074400(n) = 2*A272027(n).
a(n) = A000203(n) + A274535(n) = A074400(n) + A239050(n).
Dirichlet g.f.: 6*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016
Conjecture: a(n) = sigma(5*n) = A283118(n) iff n is not a multiple of 5. - Omar E. Pol, Oct 02 2018

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)

A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 07 2021

Conjecture: the sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

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

Columns 1, 3 and 4 give A000203.
Column 2 gives A000004.
Columns 5 and 6 gives A074400.
Column 7 and 8 give A239050.
Column 9 gives A319527.
Column 10 gives A319528.
Leading diagonal gives A002865.
Cf. A135010, A138879.

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 *)

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)

cp's OEIS Frontend

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

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

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A274536 a(n) = 6 * sigma(n).

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

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A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-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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