The Smallest Parts Partition Function

Today I turned 25773 days old and I felt it shouldn't pass without making mention of this number's connection to the Smallest Parts Partition Function. This function assigns to each natural number \(n\), another number which is the total of the smallest parts in all partitions of \(n\). Here is the mapping, from \(n=1\) up to \(n=30\):

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589, 801, 1048, 1407, 1820, 2399, 3087, 3998, 5092, 6545, 8263, 10486, 13165, 16562, 20630, 25773, ...

Figure 1 shows the SageMath code that I wrote to generate this sequence, up to and including 25773. Here is the Permalink.

Figure 1: SageMath code to generate
Smallest Parts Partition Numbers

Here is the example given in the OEIS A092269 comments:

Partitions of 4 are [1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]. 

1 appears four times in [1, 1, 1, 1]

1 appears two times in [1, 1, 2]

2 appears two times in [2, 2]

1 appears once in [1, 3]

4 appears once in [4]

Thus a(4)=4+2+2+1+1=10

Figure 2 shows a plot of the values up to 25773:

Figure 2: plot of the Smallest Parts Partition Function

Like the partition function, there is a generating function but it's rather complicated and I won't include it here. However, it can be viewed in the OEIS comments. I just wanted to mention it because the next member of the sequence is 31897 which is a long way off. There are a number of academic papers about this function so it is a topic of serious mathematical interest.

The number of partitions from 1 to 30 are shown in the list below:

[1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604]

It's interesting to look at the ratio between the total value of the number of smallest parts and the number of partitions for numbers between 1 and 30. The results and a plot of these values can be found in Figure 3.

Figure 3