Spt function

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.

The first few values of spt(n) are:


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

Example
For example, there are five partitions of 4 (with smallest parts underlined):


 * 4
 * 3 + 1
 * 2 + 2
 * 2 + 1 + 1
 * 1 + 1 + 1 + 1

These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively. So spt(4) = 1 + 1 + 2 + 2 + 4 = 10.

Properties
Like the partition function, spt(n) has a generating function. It is given by
 * $$S(q)=\sum_{n=1}^{\infty} \mathrm{spt}(n) q^n=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty} \frac{q^n \prod_{m=1}^{n-1}(1-q^m)}{1-q^n}$$

where $$(q)_{\infty}=\prod_{n=1}^{\infty} (1-q^n)$$.

The function $$S(q)$$ is related to a mock modular form. Let $$E_2(z)$$ denote the weight 2 quasi-modular Eisenstein series and let $$\eta(z)$$ denote the Dedekind eta function. Then for $$q=e^{2\pi i z}$$, the function
 * $$\tilde{S}(z):=q^{-1/24}S(q)-\frac{1}{12}\frac{E_2(z)}{\eta(z)}$$

is a mock modular form of weight 3/2 on the full modular group $$SL_2(\mathbb{Z})$$ with multiplier system $$\chi_{\eta}^{-1}$$, where $$\chi_{\eta}$$ is the multiplier system for $$\eta(z)$$.

While a closed formula is not known for spt(n), there are Ramanujan-like congruences including
 * $$\mathrm{spt}(5n+4) \equiv 0 \mod(5) $$
 * $$\mathrm{spt}(7n+5) \equiv 0 \mod(7) $$
 * $$\mathrm{spt}(13n+6) \equiv 0 \mod(13).$$