The spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each integer partition of a positive integer.
It is related to the partition function.
[1] The first few values of spt(n) are: For example, there are five partitions of 4 (with smallest parts underlined): These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively.
Like the partition function, spt(n) has a generating function.
is related to a mock modular form.
denote the weight 2 quasi-modular Eisenstein series and let
denote the Dedekind eta function.
, the function is a mock modular form of weight 3/2 on the full modular group
While a closed formula is not known for spt(n), there are Ramanujan-like congruences including
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