Sum of divisors functions written in terms of partition function

Theorem

The following formula holds: $$\sigma_1(n)=p(n)+\displaystyle\sum_{1 \leq \frac{3k^2 \pm k}{2} \leq n} (-1)^k\dfrac{3k^2 \pm k}{2} p \left(n - \dfrac{3k^2 \pm k}{2} \right),$$ where $\sigma_1$ denotes the sum of divisors function and $p$ denotes the partition function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.2.1 \mathrm{II}.B.$