A340527 -id:A340527 - cp's OEIS frontend

A182738 Partial sums of A066186.

1, 5, 14, 34, 69, 135, 240, 416, 686, 1106, 1722, 2646, 3959, 5849, 8489, 12185, 17234, 24164, 33474, 46014, 62646, 84690, 113555, 151355, 200305, 263641, 344911, 449015, 581400, 749520, 961622, 1228790, 1563509, 1982049

Offset: 1

Omar E. Pol, Jan 22 2011

a(n) is also the volume of a three-dimensional version of the section model of partitions: the 3D illustrations in A135010 show boxes with face areas of 1 X 1, 2 X 2, 3 X 3, 4 X 5, 5 X 7 units along the m and p(m) axis, which is sequence A066186. Assuming that the boxes are 1 unit deep, the total volume of all boxes up to layer n is a(n). See the first two links.
From Omar E. Pol, Jan 20 2021: (Start)
a(n) is the sum of all parts of all partitions of all positive integers <= n.
Convolution of A000203 and A000070.
Convolution of A024916 and A000041.
Convolution of A175254 and A002865.
Convolution of A340793 and A014153.
Row sums of triangles A340527, A340531, A340579.
Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593). The total area of the terraces equals A024916(n), the same as the area of the base.
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
a(n) is the volume (or the total number of unit cubes) of the polycube.
That is due to the correspondence between divisors and partitions (cf. A336811).
The symmetric tower is a member of the family of the pyramid described in A245092.
The growth of the volume of the polycube represents every convolution mentioned above. (End)

			a(6) = 135 because the volume V(6) = p(1) + 2*p(2) + 3*p(3) + 4*p(4) + 5*p(5) + 6*p(6) = 1 + 2*2 + 3*3 + 4*5 + 5*7 + 6*11 = 1 + 4 + 9 + 20 + 35 + 66 = 135 where p(n) = A000041(n).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Omar E. Pol, Illustration of a(6) = 135, the polycube contains 135 unit cubes.
Omar E. Pol, Illustration of a(9) = 686, the polycube contains 686 unit cubes.

Cf. A000041, A000070, A000203, A002865, A014153, A024916, A066186, A135010, A175254, A237270, A237593, A245092, A336811, A340527, A340531, A340579, A340793.

With[{no=35},Accumulate[PartitionsP[Range[no]]Range[no]]] (* Harvey P. Dale, Feb 02 2011 *)

a(n) = n*A000070(n) - A014153(n-1). - Vaclav Kotesovec, Jun 23 2015
a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(6)/Pi)/sqrt(n) + (73*Pi^2/6912 - 3/16)/n). - Vaclav Kotesovec, Jun 23 2015, extended Nov 04 2016
G.f.: x*f'(x)/(1 - x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017

A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.

Original entry on oeis.org

1, 4, 1, 8, 4, 1, 1, 15, 8, 4, 4, 1, 1, 1, 21, 15, 8, 8, 4, 4, 4, 1, 1, 1, 1, 1, 33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 41, 33, 21, 21, 15, 15, 15, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 41, 33, 33, 21, 21, 21, 15, 15, 15, 15, 15

Offset: 1

Omar E. Pol, Jan 10 2021

Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593).
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
T(n,k) is the volume (the number of cells) in the k-th level starting from the base.
This polycube has the property that the volume (the total number of cells) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
A dissection of the symmetric tower is a three-dimensional spiral whose top view is described in A239660.
Other triangles related to the volume of this polycube are A340527 and A340579.
The symmetric tower is a member of the family of the stepped pyramid described in A245092.
For another symmetric tower of the same family and whose volume equals A066186(n) see A340423.
The sum of row n of triangle equals A182738(n). That property is due to the correspondence between divisors and parts. For more information see A336811.

			Triangle begins:
   1;
   4,  1;
   8,  4,  1,  1;
  15,  8,  4,  4, 1, 1, 1;
  21, 15,  8,  8, 4, 4, 4, 1, 1, 1, 1, 1;
  33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 5 the length of row 5 is A000070(4) = 12.
The sum of row 5 is 21 + 15 + 8 + 8 + 4 + 4 + 4 + 1 + 1 + 1 + 1 + 1 = 69, equaling A182738(5).

Row sums give A182738.
Cf. A340527 (a regular version).
Members of the same family are: A176206, A337209, A339258, A340530.
Cf. A221529, A239660, A339278, A339304, A340423, A340529.
Cf. A000070, A024916, A237593, A336811, A338156.

a(m) = A024916(A176206(m)), assuming A176206 has offset 1.

T(n,k) = A024916(A176206(n,k)), assuming A176206 has offset 1.

A340526 Triangle read by rows: T(n,k) = A006218(n-k+1)*A000041(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 5, 3, 2, 8, 5, 6, 3, 10, 8, 10, 9, 5, 14, 10, 16, 15, 15, 7, 16, 14, 20, 24, 25, 21, 11, 20, 16, 28, 30, 40, 35, 33, 15, 23, 20, 32, 42, 50, 56, 55, 45, 22, 27, 23, 40, 48, 70, 70, 88, 75, 66, 30, 29, 27, 46, 60, 80, 98, 110, 120, 110, 90, 42, 35, 29, 54, 69, 100, 112, 154, 150, 176, 150, 126, 56

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture 1: T(n,k) is the total number of divisors of the terms that are in the k-th blocks of the first n rows of triangle A176206.
Conjecture 2: the sum of row n equals A284870, the total number of parts in all partitions of all positive integers <= n.
The above conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).

			Triangle begins:
   1;
   3,  1;
   5,  3,  2;
   8,  5,  6,  3;
  10,  8, 10,  9,   5;
  14, 10, 16, 15,  15,   7;
  16, 14, 20, 24,  25,  21,  11;
  20, 16, 28, 30,  40,  35,  33,  15;
  23, 20, 32, 42,  50,  56,  55,  45,  22;
  27, 23, 40, 48,  70,  70,  88,  75,  66,  30;
  29, 27, 46, 60,  80,  98, 110, 120, 110,  90,  42;
  35, 29, 54, 69, 100, 112, 154, 150, 176, 150, 126,  56;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000041         T(6,k)
1      1  *  14   =   14
2      1  *  10   =   10
3      2  *   8   =   16
4      3  *   5   =   15
5      5  *   3   =   15
6      7  *   1   =    7
.          A006218
--------------------------
The sum of row 6 is 14 + 10 + 16 + 15 + 15 + 7 = 77, equaling A284870(6).

Columns 1 and 2 give A006218.
Leading diagonal gives A000041.
Row sums give A284870.
Cf. A176206, A221531, A339106, A340424, A340425, A340524, A340426, A340527, A336811.

f(n) = sum(k=1, n, n\k); \\ A006218
T(n,k) = f(n-k+1)*numbpart(k-1); \\ Michel Marcus, Jan 15 2021

A340579 Triangle read by rows: T(n,k) = A000203(n-k+1)*A000070(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 4, 6, 4, 7, 8, 12, 7, 6, 14, 16, 21, 12, 12, 12, 28, 28, 36, 19, 8, 24, 24, 49, 48, 57, 30, 15, 16, 48, 42, 84, 76, 90, 45, 13, 30, 32, 84, 72, 133, 120, 135, 67, 18, 26, 60, 56, 144, 114, 210, 180, 201, 97, 12, 36, 52, 105, 96, 228, 180, 315, 268, 291, 139, 28, 24, 72, 91

Offset: 1

Omar E. Pol, Jan 12 2021

Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593). The total area of the terraces equals A024916(n), the same as the area of the base.
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
T(n,k) is the total volume (or total number of cubes) exactly below the symmetric representation of sigma(n-k+1). In other words: T(n,k) is the total volume (the total number of cubes) exactly below the terraces that are in the k-th level that contains terraces starting from the base.
This symmetric tower has the property that its volume (the total number of cubes) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n. That is due to the correspondence between divisors and partitions (cf. A336811).
The growth of the volume represents the convolution of A000203 and A000070.
The symmetric tower is a member of the family of the pyramid described in A245092.
For another symmetric tower of the same family and whose volume equals A066186(n) see A221529 and A339106.

			Triangle begins:
   1;
   3,   2;
   4,   6,   4;
   7,   8,  12,   7;
   6,  14,  16,  21,  12;
  12,  12,  28,  28,  36,  19;
   8,  24,  24,  49,  48,  57,  30;
  15,  16,  48,  42,  84,  76,  90,  45;
  13,  30,  32,  84,  72, 133, 120, 135,  67;
  18,  26,  60,  56, 144, 114, 210, 180, 201,  97;
  12,  36,  52, 105,  96, 228, 180, 315, 268, 291, 139;
...
For n = 6 the calculation of every term of row 6 is as follows:
-------------------------
k   A000070        T(6,k)
1      1  *  12  =   12
2      2  *  6   =   12
3      4  *  7   =   28
4      7  *  4   =   28
5     12  *  3   =   36
6     19  *  1   =   19
.         A000203
-------------------------
The sum of row 6 is 12 + 12 + 28 + 28 + 36 + 19 = 135, equaling A182738(6).

Row sums give A182738.

Cf. A000070, A000203, A024916, A221529, A221531, A237593, A339106, A340424, A340426, A340524, A340525, A340526, A340527, A340531.

row(n) = vector(n, k, sigma(n-k+1)*sum(i=0, k-1, numbpart(i))); \\ Michel Marcus, Jul 23 2021

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A182738 Partial sums of A066186.

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A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.

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A340526 Triangle read by rows: T(n,k) = A006218(n-k+1)*A000041(k-1), 1 <= k <= n.

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A340579 Triangle read by rows: T(n,k) = A000203(n-k+1)*A000070(k-1), 1 <= k <= n.

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