A141285 -id:A141285 - cp's OEIS frontend

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

1, 4, 7, 10, 16, 22, 34, 46, 67, 91, 127, 169, 232, 304, 406, 529, 694, 892, 1156, 1471, 1882, 2377, 3007, 3766, 4726, 5875, 7309, 9031, 11155, 13696, 16813, 20527, 25048, 30430, 36931, 44650, 53932, 64912, 78046, 93556, 112015, 133750, 159523, 189784, 225526, 267403, 316675, 374263, 441820, 520576, 612679

Offset: 0

Omar E. Pol, Feb 11 2018

nonn

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

			Construction of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     7      10        16          22            34     (vertices)
A299474(n):   4     8      12        20          28            44     (edges)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.

a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.

a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

A179862 An unrestricted partition statistic: sum of A179864 over row n.

Original entry on oeis.org

1, 4, 9, 19, 33, 59, 93, 150, 226, 342, 494, 721, 1011, 1425, 1960, 2695, 3633, 4903, 6506, 8633, 11312, 14796, 19157, 24773, 31744, 40608, 51578, 65372, 82341, 103522, 129428, 161505, 200589, 248614, 306869, 378051, 463987, 568387, 693989, 845754, 1027625

Offset: 1

Alford Arnold, Aug 02 2010

nonn

Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n minus the number of partitions of n. - Omar E. Pol, Jul 15 2013

Sum of the hook-lengths of the (1,1)-cells of the Ferrers diagrams over all partitions of n. Example: a(3) = 9 because in each of the partitions 3, 21, and 111 the (1,1)-cell has hook-length 3. Comment follows at once from the previous comment. - Emeric Deutsch, Dec 20 2015

			From _Omar E. Pol_, Jul 15 2013: (Start)
Illustration of initial terms using a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). a(n) is the x-coordinate of the mentioned largest peak. Note that this Dyck path is infinite.
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = this sequence (End)

Cf. A179864.

a(n) = Sum_{k=1..A000041(n)} A179864(n,k).
a(n) = A211978(n) - A000041(n). - Omar E. Pol, Jul 15 2013
a(n) = A225600(A139582(n)-1), n>= 1. - Omar E. Pol, Jul 25 2013

More terms from Omar E. Pol, Jul 15 2013

A182727 Sum of largest parts of the shell model of partitions with n regions.

Original entry on oeis.org

1, 3, 6, 8, 12, 15, 20, 22, 26, 29, 35, 38, 43, 47, 54, 56, 60, 63, 69, 74, 78, 86, 89, 94, 98, 105, 108, 114, 119, 128, 130, 134, 137, 143, 148, 152, 160, 164, 171, 177, 182, 192, 195, 200, 204, 211, 214, 220, 225, 234, 239, 243, 251, 258, 264, 275, 277, 281

Offset: 1

Omar E. Pol, Jan 25 2011

Question: Is there some connection with fractals?

			For n = 6 the largest parts of the first six regions of the shell model of partitions are 1, 2, 3, 2, 4, 3, so a(6) = 1+2+3+2+4+3 = 15.
Written as a triangle begins:
1;
3;
6;
8,   12;
15,  20;
22,  26, 29, 35;
38,  43, 47, 54;
56,  60, 63, 69, 74, 78, 86;
89,  94, 98,105,108,114,119,128;
130,134,137,143,148,152,160,164,171,177,182,192;
195,200,204,211,214,220,225,234,239,243,251,258,264,275;

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.

Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

a(A000041(n)) = A182181(A000041(n)) = A006128(n). - Omar E. Pol, May 24 2012

New name from Omar E. Pol, Apr 26 2012

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 17, 23, 33, 46, 64, 86, 121, 161, 217, 291, 388, 507, 671, 870, 1131, 1458, 1872, 2383, 3042, 3840, 4841, 6076, 7605, 9460, 11765, 14544, 17950, 22073, 27077, 33092, 40395, 49113, 59611, 72162, 87185, 105035, 126366

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The physical model shows each part represented by an object, for example using a cube or a cuboid. In this case the small version of the model shows each part as a cube of side 1 which is labeled with the size of the part. On the same way the large version of the model shows each part as a cuboid of sides 1 x 1 x L where L is the size of the part. The cuboid is labeled with the level of the part. For the sum of parts see A194804. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
1) Small version:
.
Level
1        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
2                     3 2 2 1 2 2 3
3                         2 1 2
4                           1
5      Table 2.0            1            Table 2.1
6                           1
7                           1
.
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
2) Large version:
.
.                   . . . . 1 . . . .
.                   . . . 1 2 . . . .
.                   . 1 . . 2 1 . . .
.                   . . 1 2 2 . . 1 .
.                   . . . . 2 2 1 . .
.                   1 2 2 3 2 . . . .
.                           2 3 2 2 1
.
The large version shows the parts labeled with the level of the part where "the level of a part" is its position in the partition. In both versions there are 23 parts that are visible, so a(7) = 23. Also using the formula we have a(7) = 7+8+8 = 23.

Cf. A006128, A096541, A138135, A135010, A138121, A141285, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194804, A194805, A210979.

a(n) = n + A138135(n-1) + A138135(n), if n >= 2.

A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 2, 3, 6, 2, 2, 3, 7, 2, 2, 2, 2, 3, 4, 8, 2, 2, 2, 2, 3, 3, 4, 9, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 10, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6, 13

Offset: 1

Omar E. Pol, Oct 21 2011

For n >= 2, row n lists the parts of the head of the last section of the set of partitions of n, except the emergent parts.

Also 1 together with the integers > 1 of A196931.

			Written as a triangle:
1,
2,
3,
2,4,
2,5,
2,2,3,6
2,2,3,7,
2,2,2,2,3,4,8,
2,2,2,2,3,3,4,9,
2,2,2,2,2,2,2,3,3,4,5,10,
2,2,2,2,2,2,2,2,3,3,3,4,5,11,
2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12,
2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13,
...
Row n has length A002865(n), n >= 2. The sum of row n is A182708(n), n >= 2. The number of 2's in row n is A002865(n-2), n >= 4. Right border of triangle gives A000027.

Alois P. Heinz, Rows n = 1..33, flattened

Where records occur give A000041.

Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931.

p:= (f, g)-> zip((x, y)->x+y, f, g, 0):
b:= proc(n, i) option remember; local g, j, r;
      if n=0 then [1] elif i<2 then [0]
    else r:= b(n, i-1);
         for j to n/i do g:= b(n-i*j, i-1);
           r:= p(p(r, [0$i, g[1]]), subsop(1=0, g));
         od; r
      fi
    end:
T:= proc(n) local l; l:= b(n$2);
      `if`(n=1, 1, seq(i$l[i+1], i=2..nops(l)-1))
    end:
seq(T(n), n=1..16);  # Alois P. Heinz, May 30 2013

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{ g, j, r}, Which[n == 0, {1}, i<2, {0}, True, r = b[n, i-1]; For[j = 1, j <= n/i, j++, g = b[n-i*j, i-1]; r = p[p[r, Append[Array[0&, i], g // First]], ReplacePart[g, 1 -> 0]]]; r]]; T[n_] := Module[{l}, l = b[n, n]; If[n == 1, {1}, Table[Array[i&, l[[i+1]]], {i, 2, Length[l]-1}] // Flatten]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".

Original entry on oeis.org

1, 5, 11, 27, 43, 93, 131, 247, 352, 584, 808, 1306, 1735, 2643, 3568, 5160, 6835, 9721, 12672, 17564, 22832, 30818, 39743, 53027, 67594, 88740, 112752, 145944, 183979, 236059, 295370, 375208, 467363, 588007, 728437, 910339, 1121009, 1391083, 1706003, 2103013

Offset: 1

Omar E. Pol, Feb 08 2012

nonn

Since partial sums of this sequence give A066183 we have that A066183(n) is also the volume of the mentioned version of the shell model of partitions with n shells. Each part of size k has a volume equal to k^2 since each box is a cuboid whose sides have lengths: 1, k, k.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Row sums of triangle A206438. Partial sums give A066183.

Cf. A135010, A138121, A141285.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1)+(p-> p+[0, p[1]*i^2])(b(n-i, min(n-i, i))))
    end:
a:= n-> (b(n$2)-b(n-1$2))[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 23 2022

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n},
     b[n, i-1] + Function[p, p + {0, p[[1]]*i^2}][b[n-i, Min[n-i, i]]]];
a[n_] := (b[n, n] - b[n-1, n-1])[[2]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 25 2022, after Alois P. Heinz *)

a(n) ~ sqrt(3) * zeta(3) * exp(Pi*sqrt(2*n/3)) / Pi^2. - Vaclav Kotesovec, Oct 20 2024

A210941 Triangle read by rows in which row n lists the parts > 1 of the n-th zone of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 2, 4, 3, 2, 5, 2, 2, 2, 4, 2, 3, 3, 6, 3, 2, 2, 5, 2, 4, 3, 7, 2, 2, 2, 2, 4, 2, 2, 3, 3, 2, 6, 2, 5, 3, 4, 4, 8, 3, 2, 2, 2, 5, 2, 2, 4, 3, 2, 7, 2, 3, 3, 3, 6, 3, 5, 4, 9, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 5, 3, 2, 4, 4, 2, 8, 2

Offset: 1

Omar E. Pol, Apr 18 2012

The n-th zone of the shell model of partitions is formed by the parts of row n followed by infinitely many parts of size 1 (see example and also A210943).

Row n lists the largest part and the parts > 1 of the n-th zone of the model.

			Triangle                First 15 zones of the
begins                  shell model of partitions
--------------------------------------------------
1;                      1 1 1 1 1 1 1 1 1 1 1...
2;                      . 2 1 1 1 1 1 1 1 1 1...
3;                      . . 3 1 1 1 1 1 1 1 1...
2, 2;                   . 2 . 2 1 1 1 1 1 1 1...
4;                      . . . 4 1 1 1 1 1 1 1...
3, 2;                   . . 3 . 2 1 1 1 1 1 1...
5;                      . . . . 5 1 1 1 1 1 1...
2, 2, 2;                . 2 . 2 . 2 1 1 1 1 1...
4, 2;                   . . . 4 . 2 1 1 1 1 1...
3, 3;                   . . 3 . . 3 1 1 1 1 1...
6;                      . . . . . 6 1 1 1 1 1...
3, 2, 2;                . . 3 . 2 . 2 1 1 1 1...
5, 2;                   . . . . 5 . 2 1 1 1 1...
4, 3;                   . . . 4 . . 3 1 1 1 1...
7;                      . . . . . . 7 1 1 1 1...

Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12
Mikhail Kurkov, PARI program. [verification needed]

Column 1 is A141285. Row n has length A194548(n), n > 1.

Cf. A135010, A138121, A182703, A194711, A206437, A210942, A210943.

a210941(n)={
    my(p=[],r=[1]);
    if(n>1,
    my(c=2);
    while(#r1]));
        c++));
    return(r[1..n])
} \\ Joe Slater, Sep 02 2024

A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 2, 3, 9, 7, 48, 15, 119, 72, 269, 56, 2740, 101, 1163, 1208, 5218, 297, 24319, 490, 42150, 6669, 14098, 1255, 792335, 5564, 42501, 30585, 432413, 4565, 4513067, 6842, 1251217, 122818, 317297, 124253, 54782479, 21637, 802541, 445414, 48590725, 44583

Offset: 1

Omar E. Pol, Mar 25 2018

nonn

If n is a noncomposite number (that is, 1 or prime), then a(n) = A000041(n).

For n >= 3, p(sigma(n-2)) < a(n) <= p(sigma(n-1)), where p(n) = A000041(n) and sigma(n) = A000203(n).

			For n = 4 the sum of the divisors of 4 is 1 + 2 + 4 = 7. Then we have that, in list of colexicographically ordered partitions of 7, the divisors of 4 are in the 9th partition, so a(4) = 9 (see below):
------------------------------------------------------
   k        Diagram        Partitions of 7
------------------------------------------------------
         _ _ _ _ _ _ _
   1    |_| | | | | | |    [1, 1, 1, 1, 1, 1, 1]
   2    |_ _| | | | | |    [2, 1, 1, 1, 1, 1]
   3    |_ _ _| | | | |    [3, 1, 1, 1, 1]
   4    |_ _|   | | | |    [2, 2, 1, 1, 1]
   5    |_ _ _ _| | | |    [4, 1, 1, 1]
   6    |_ _ _|   | | |    [3, 2, 1, 1]
   7    |_ _ _ _ _| | |    [5, 1, 1]
   8    |_ _|   |   | |    [2, 2, 2, 1]
   9    |_ _ _ _|   | |    [4, 2, 1]       <---- Divisors of 4
  10    |_ _ _|     | |    [3, 3, 1]
  11    |_ _ _ _ _ _| |    [6, 1]
  12    |_ _ _|   |   |    [3, 2, 2]
  13    |_ _ _ _ _|   |    [5, 2]
  14    |_ _ _ _|     |    [4, 3]
  15    |_ _ _ _ _ _ _|    [7]
.

Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..700 from Andrew Howroyd)

Cf. A000040, A000041, A000203, A008578, A026807, A027750, A056538, A135010, A141285, A194446, A211992, A272024.

b[n_, k_] := b[n, k] = If[k < 1 || k > n, 0, If[n == k, 1, b[n, k + 1] + b[n - k, k]]];
PartIndex[v_] := Module[{s = 1, t = 0}, For[i = Length[v], i >= 1, i--, t += v[[i]]; s += b[t, If[i == 1, 1, v[[i - 1]]]] - b[t, v[[i]]]]; s];
a[n_] := PartIndex[Divisors[n]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 27 2019, after Andrew Howroyd *)

a(n)={my(d=divisors(n)); vecsearch(vecsort(partitions(vecsum(d))), d)} \\ Andrew Howroyd, Jul 15 2018

\\ here b(n,k) is A026807.
b(n,k)=polcoeff(1/prod(i=k, n, 1-x^i + O(x*x^n)), n)
PartIndex(v)={my(s=1,t=0); forstep(i=#v, 1, -1, t+=v[i]; s+=b(t, if(i==1, 1, v[i-1])) - b(t, v[i])); s}
a(n)=PartIndex(divisors(n)); \\ Andrew Howroyd, Jul 15 2018

Terms a(8) and beyond from Andrew Howroyd, Jul 15 2018

A194797 Imbalance of the sum of parts of all partitions of n.

Original entry on oeis.org

0, -2, 1, -7, 3, -21, 7, -49, 23, -97, 57, -195, 117, -359, 256, -624, 498, -1086, 909, -1831, 1634, -2986, 2833, -4847, 4728, -7700, 7798, -12026, 12537, -18633, 19745, -28479, 30723, -42955, 47100, -64284, 71136, -95228, 106402, -139718, 157327, -203495

Offset: 1

Omar E. Pol, Jan 31 2012

sign

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see example). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. Note that every column contains exactly the same parts. For more information see A135010.

			For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
------------------------------------------------------
Partitions                Tree             Table 1.0
of 6.                    A194805            A135010
------------------------------------------------------
6                   6                     6 . . . . .
3+3                   3                   3 . . 3 . .
4+2                     4                 4 . . . 2 .
2+2+2                     2               2 . 2 . 2 .
5+1                         1   5         5 . . . . 1
3+2+1                       1 3           3 . . 2 . 1
4+1+1                   4   1             4 . . . 1 1
2+2+1+1                   2 1             2 . 2 . 1 1
3+1+1+1                     1 3           3 . . 1 1 1
2+1+1+1+1                 2 1             2 . 1 1 1 1
1+1+1+1+1+1                 1             1 1 1 1 1 1
------------------------------------------------------
.
.                   6 3 4 2 1 3 5
.     Table 2.0     . . . . 1 . .     Table 2.1
.      A182982      . . . 2 1 . .      A182983
.                   . 3 . . 1 2 .
.                   . . 2 2 1 . .
.                   . . . . 1
------------------------------------------------------
The sum of all parts > 1 on the left hand side is 34 and the sum of all parts > 1 on the right hand side is 13, so a(6) = -34 + 13 = -21. On the other hand for n = 6 the first n terms of A138880 are 0, 2, 3, 8, 10, 24 thus a(6) = 0-2+3-8+10-24 = -21.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A002865, A135010, A138121, A138880, A141285, A182710, A182742, A182743, A182746, A182747, A182982, A182983, A182994, A182995, A194796, A194805.

with(combinat):
a:= proc(n) option remember;
      n *(-1)^n *(numbpart(n-1)-numbpart(n)) +a(n-1)
    end: a(0):=0:
seq(a(n), n=1..50); # Alois P. Heinz, Apr 04 2012

a[n_] := Sum[(-1)^(k-1)*k*(PartitionsP[k] - PartitionsP[k-1]), {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Dec 09 2016 *)

a(n) = Sum_{k=1..n} (-1)^(k-1)*k*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
a(n) = b(1)-b(2)+b(3)-b(4)+b(5)-b(6)...+-b(n), where b(n) = A138880(n).
a(n) ~ -(-1)^n * Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (48*sqrt(n)). - Vaclav Kotesovec, Oct 09 2018

A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 46, 59, 74, 90, 109, 129, 151, 174, 201, 229, 259, 290, 323, 358, 394, 434, 475, 518, 562, 609, 657, 707, 758, 814, 871, 930, 990, 1052, 1116, 1181, 1249, 1318, 1389, 1462, 1536, 1615, 1695, 1777, 1860, 1946, 2033, 2122, 2212, 2305, 2400, 2496, 2594, 2694, 2795

Offset: 1

Omar E. Pol, May 16 2016

nonn

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(n) and the n-th vertical line segment has length A194446(n). Both line segments shares the point (A141285(n),n). Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first k sections from the table we have a representation of the partitions of k. For the definition of "region" see A206437. For the definition of "section" see A135010.

			For n = 4 the corner of size 4 X 4 of the modular table of partitions contains 11 parts as shown below, so a(4) = 11.
.
.   Row   _ _ _ _       Parts
.    1   |_| | | |        4
.    2   |_ _| | |        3
.    3   |_ _ _| |        2
.    4   |_ _|   |        2
.                       ----
.                  Total 11
.
For n = 20 the corner of size 20 X 20 of the modular table of partitions contains 323 parts as shown below, so a(20) = 323.
.
.   Row   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _       Parts
.    1   |_| | | | | | | | | | | | | | | | | | | |        20
.    2   |_ _| | | | | | | | | | | | | | | | | | |        19
.    3   |_ _ _| | | | | | | | | | | | | | | | | |        18
.    4   |_ _|   | | | | | | | | | | | | | | | | |        18
.    5   |_ _ _ _| | | | | | | | | | | | | | | | |        17
.    6   |_ _ _|   | | | | | | | | | | | | | | | |        17
.    7   |_ _ _ _ _| | | | | | | | | | | | | | | |        16
.    8   |_ _|   |   | | | | | | | | | | | | | | |        17
.    9   |_ _ _ _|   | | | | | | | | | | | | | | |        16
.   10   |_ _ _|     | | | | | | | | | | | | | | |        16
.   11   |_ _ _ _ _ _| | | | | | | | | | | | | | |        15
.   12   |_ _ _|   |   | | | | | | | | | | | | | |        16
.   13   |_ _ _ _ _|   | | | | | | | | | | | | | |        15
.   14   |_ _ _ _|     | | | | | | | | | | | | | |        15
.   15   |_ _ _ _ _ _ _| | | | | | | | | | | | | |        14
.   16   |_ _|   |   |   | | | | | | | | | | | | |        16
.   17   |_ _ _ _|   |   | | | | | | | | | | | | |        15
.   18   |_ _ _|     |   | | | | | | | | | | | | |        15
.   19   |_ _ _ _ _ _|   | | | | | | | | | | | | |        14
.   20   |_ _ _ _ _|     | | | | | | | | | | | | |        14
.                                                       -----
.                                                  Total 323
.

Cf. A000041, A006128, A135010, A141285, A194446, A206437, A211992, A220482, A228525, A278602.

cp's OEIS Frontend

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A179862 An unrestricted partition statistic: sum of A179864 over row n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A182727 Sum of largest parts of the shell model of partitions with n regions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A210941 Triangle read by rows in which row n lists the parts > 1 of the n-th zone of the shell model of partitions, with a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs