The spt function is a function in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 that counts the sum of the number of smallest parts in each partition

Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...

 of a positive integer. It is related to the partition function.

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.

The first few values of spt(n) are:

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ...

Like many functions in mathematics, spt(n) has a generating function

Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...

. There are connections to Maass forms, and under certain conditions the generating function is an eigenform

Eigenform

An eigenform is a modular form which is an eigenvector for all Hecke operators Tm, m = 1, 2, 3, …....

 for some Hecke operator

Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....

s.

While a closed formula is not known for spt(n), there are Ramanajuan-like congruences

Ramanujan's congruences

In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p. The Indian mathematician Srinivasa Ramanujan discovered the following*p\equiv 0 \pmod 5*p\equiv 0 \pmod 7...

including