Basic Examples

Here is an illustrative example:
[◼]
SmallestPartsFunction
[3]
5
Out[]=
The integer partitions of 3:
IntegerPartitions@3
In[]:=
{{3},{2,1},{1,1,1}}
Out[]=
The smallest parts:
Min/@IntegerPartitions@3
In[]:=
{3,1,1}
Out[]=
The number of smallest parts per partition:
Count[#,Min[#]]&/@IntegerPartitions@3
In[]:=
{1,1,3}
Out[]=
Their total:
Total@%
In[]:=
5
Out[]=
Here are the first 10 numbers in the smallest part function
(spt
) sequence:
[◼]
SmallestPartsFunction
/@Range@10
{1,3,5,10,14,26,35,57,80,119}
Out[]=

Neat Examples

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:
Table
[◼]
SmallestPartsFunction
[5n+4],{n,10}
{80,440,1820,6545,20630,59960,161840,412950,1002435,2335760}
Out[]=
Numbers of the form
7n+5
have
spt
divisible by 7:
1/7Table
[◼]
SmallestPartsFunction
[7n+5],{n,7}
{34,260,1498,6956,28024,100953,333680}
Out[]=
Numbers of the form
13n+6
have
spt
divisible by 13:
1/13Table
[◼]
SmallestPartsFunction
[13n+6],{n,4}
{140,3042,38054,344212}
Out[]=