A182709 -id:A182709 - cp's OEIS frontend

A220489 Total number of smallest parts in all partitions of n minus the total number of smallest parts that are also emergent parts in all partitions of n with at least one distinct part.

1, 3, 5, 10, 14, 26, 34, 56, 77, 114, 153, 227, 296, 414, 555, 750, 981, 1316, 1702, 2241, 2887, 3727, 4761, 6112, 7725, 9787, 12316, 15473, 19307, 24099, 29867, 37004, 45626, 56147, 68856, 84297, 102793, 125167, 151969, 184166, 222553, 268529, 323152

Offset: 1

Omar E. Pol, Feb 24 2013

nonn

First differs from A092269 at a(7).

For the definition of "emergent part" see A182699, A182709.

Cf. A000005, A000041, A000070, A000203, A002865, A092269, A120452, A195820, A220479, A220483.

a(n) = spt(n) - A220483(n) = 1 + sigma(n) + A000070(n) - p(n-1) - d(n) - n.

a(n) = A092269(n) - A220483(n) = 1 + A000203(n) + A000070(n) - A000041(n-1) - A000005(n) - n.

A330242 Sum of largest emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 9, 12, 24, 33, 54, 72, 112, 144, 210, 273, 379, 485, 661, 835, 1112, 1401, 1825, 2284, 2944, 3652, 4645, 5745, 7223, 8879, 11080, 13541, 16760, 20406, 25062, 30379, 37102, 44761, 54351, 65347, 78919, 94517, 113645, 135603, 162331, 193088, 230182, 272916, 324195, 383169, 453571

Offset: 1

Omar E. Pol, Dec 06 2019

nonn

In other words: a(n) is the sum of the largest parts of all partitions of n that contain emergent parts.
The partitions of n that contain emergent parts are the partitions that contain neither 1 nor n as a part. All parts of these partitions are emergent parts except the last part of every partition.
For the definition of emergent part see A182699.

			For n = 9 the diagram of
the partitions of 9 that
do not contain 1 as a part
is as shown below:           Partitions
.
    |_ _ _|   |   |   |      [3, 2, 2, 2]
    |_ _ _ _ _|   |   |      [5, 2, 2]
    |_ _ _ _|     |   |      [4, 3, 2]
    |_ _ _ _ _ _ _|   |      [7, 2]
    |_ _ _|     |     |      [3, 3, 3]
    |_ _ _ _ _ _|     |      [6, 3]
    |_ _ _ _ _|       |      [5, 4]
    |_ _ _ _ _ _ _ _ _|      [9]
.
Note that the above diagram is also the "head" of the last section of the set of partitions of 9, where the "tail" is formed by A000041(9-1)= 22 1's.
The diagram of the
emergent parts is as
shown below:                 Emergent parts
.
    |_ _ _|   |   |          [3, 2, 2]
    |_ _ _ _ _|   |          [5, 2]
    |_ _ _ _|     |          [4, 3]
    |_ _ _ _ _ _ _|          [7]
    |_ _ _|     |            [3, 3]
    |_ _ _ _ _ _|            [6]
    |_ _ _ _ _|              [5]
.
The sum of the largest emergent parts is 3 + 5 + 4 + 7 + 3 + 6 + 5 = 33, so a(9) = 33.

Cf. A000041, A002865, A006128, A135010, A138135, A138137, A141285, A182699, A182703, A182709, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A207031, A299474, A299475.

a(n) = A138137(n) - n.

a(n) = A207031(n,1) - n.

cp's OEIS Frontend

A220489 Total number of smallest parts in all partitions of n minus the total number of smallest parts that are also emergent parts in all partitions of n with at least one distinct part.

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