practical number

The second definition of a practical number is: a positive integer $n$ such that each smaller integer $m$ can be represented as a sum of distinct proper divisors $d_i$ of $n$ (for , with $\tau (n)$ being the divisor function; 1 is not considered a proper divisor for this application). The first few are 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, etc., listed in A007620 of Sloane’s OEIS.

For example, 12 is practical. It has for divisors 1, 2, 3, 4, 6 and 12, but only 2, 3, 4 and 6 are considered proper divisors here. The sum can consist of a single summand, so we need only concern ourselves with numbers less than 12 that are not divisors of 12. We verify that indeed 2 + 3 = 5, 3 + 4 = 7, 2 + 6 = 8, 3 + 6 = 9, 4 + 6 = 10 and 2 + 3 + 6 = 11.

Under this definition, the powers of 2 are not practical numbers. Representing odd numbers smaller than a power of 2 requires using 1 in the sums of divisors.