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 ...

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

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