The spt function (smallest parts 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(n) are:
For example, there are five partitions of 4: (1,1,1,1), (1,1,2), (1,3), (2,2) and (4). These partitions have 4, 2, 1, 2 and 1 smallest parts respectively. So spt(4) = 4 + 2 + 1 + 2 + 1 = 10.
Like the partition function, spt(n) has a generating function. It is given by
While a closed formula is not known for spt(n), there are Ramanajuan-like congruences including
- Frank Garvan. Congruences for Andrews’ spt-function modulo 32760 and extension of Atkin’s Hecke-type partition congruences.
- George Andrews. The number of smallest parts in the partitions of n.
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