Spt function
The spt function is a function in number theory that counts the sum of the number of smallest parts in each partition of a positive integer. It is related to the partition function.
The first few values of spt are:Example
For example, there are five partitions of 4 :
These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively. So spt = 1 + 1 + 2 + 2 + 4 = 10.Properties
Like the partition function, spt has a generating function. It is given by
where.
The function is related to a mock modular form. Let denote the weight 2 quasi-modular Eisenstein series and let denote the Dedekind eta function. Then for, the function
is a mock modular form of weight 3/2 on the full modular group with multiplier system, where is the multiplier system for.
While a closed formula is not known for spt, there are Ramanujan-like congruences including
