Function Repository Resource:

# SmallestPartsFunction

Total number of smallest parts in the partitions of a positive integer

Contributed by: George Beck
 ResourceFunction["SmallestPartsFunction"][n] gives the total number of smallest parts in the integer partitions of the positive integer n.

## Details and Options

A partition of n is a list of weakly decreasing positive integers that add up to n. For instance, written compactly, these are the five partitions of 4: 4, 31, 22, 211, 1111.
ResourceFunction["SmallestPartsFunction"][n] returns the Total of all the minimal elements in all the partitions of n.

## Examples

### Basic Examples (2)

Here is an illustrative example. Start with the integer partitions of 3:

 In[1]:=
 Out[1]=

Here are their smallest parts:

 In[2]:=
 Out[2]=

Compute the number of smallest parts per partition:

 In[3]:=
 Out[3]=

Find their total:

 In[4]:=
 Out[4]=

This is precisely the value returned by SmallestPartsFunction:

 In[5]:=
 Out[5]=

Here are the first 10 numbers in the smallest part function (denoted spt) sequence (OEIS A092269):

 In[6]:=
 Out[6]=

### Neat Examples (3)

The function spt(n) satisfies some congruences similar to Ramanujan’s congruences for the partition function p(n).

Numbers of the form 5n+4 have spt divisible by 5:

 In[7]:=
 Out[7]=
 In[8]:=
 Out[8]=

Numbers of the form 7n+5 have spt divisible by 7:

 In[9]:=
 Out[9]=
 In[10]:=
 Out[10]=

Numbers of the form 13n+6 have spt divisible by 13:

 In[11]:=
 Out[11]=
 In[12]:=
 Out[12]=

George Beck

## Version History

• 1.0.0 – 31 May 2019