Project Euler Problem 722
For a non-negative integer k, define where sigmak(n) = sum{d mid n} d^k is the sum of the k-th powers of the positive di
Solution
Answer: 3.376792776502e132
Using the Lambert series identity
$$E_k(q)=\sum_{n=1}^{\infty}\sigma_k(n)q^n =\sum_{d=1}^{\infty}\frac{d^k q^d}{1-q^d},$$
with
$$q = 1-\frac1{2^{25}},$$
we write $q=e^{-\varepsilon}$ where
$$\varepsilon=-\log!\left(1-\frac1{2^{25}}\right).$$
For very small $\varepsilon$, the dominant asymptotic term is
$$E_k(e^{-\varepsilon}) \sim \Gamma(k+1)\zeta(k+1)\varepsilon^{-(k+1)}.$$
For $k=15$, this approximation is extraordinarily accurate (the relative error is far below the required $10^{-12}$ precision; this is confirmed by the provided sample cases such as $k=7$, where it already matches to the shown precision).
Evaluating at
$$\varepsilon=-\log!\left(1-2^{-25}\right)$$
gives
$$E_{15}!\left(1-\frac1{2^{25}}\right) = 3.376792776502\times 10^{132}$$
rounded to twelve digits after the decimal point.
Answer: 3.376792776502e132