A practical number is a positive integer 
such that every positive
integer 
is the sum of distinct proper divisors of 
. Practical numbers were introduced by
Srinivasan (1948).
For 
with prime factorization 
, where 
, a characterization due independently
to Stewart (1954) and Sierpiński (1955) states that 
is practical if and only if, for each 
, 2, ..., 
,
 |
where 
is the divisor function and the product is defined
to be 1 when 
.
In particular, 
,
so 1 is the only odd practical number.
All even perfect numbers are practical. The number
 |
is practical for all 
,
3, .... The first few practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28,
30, 32, 36, 40, 42, 48, 54, 56, ... (OEIS A005153).
G. Melfi has computed twins, triplets, and 5-tuples of practical numbers. The
first few 5-tuples are 12, 18, 30, 198, 306, 462, 1482, 2550, 4422, ....
