Erdős-Nicolas numbers

An Erdős-Nicolas number is an integer $n$ such that the sum of its first $k$ consecutive divisors is equal to itself, with $k<\tau (n)-1$. The first ten Erdős-Nicolas numbers are 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800, 91963648 (see A194472 for more). For example, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24, and we see that 1 + 2 + 3 + 4 + 6 + 8 = 24.

Obviously a deficient number can't be an Erdős-Nicolas number. But neither can perfect numbers nor weird numbers.

J.-M. De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009): p. 141, entry for 2016