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Panarithmic number)
A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n.
The sequence of practical numbers begins
- 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 52, 54, ...
A positive integer <math>n=p_1^{\alpha_1}...p_k^{\alpha_k}<math> with <math>n>1<math> and <math>p_1perfect numbers is also a practical number.
The interest of practical numbers is that many of its properties are similar to properties of the set of prime numbers. For example, if <math>p(x)<math> is the enumerating function of practical numbers, i.e., the number of practical numbers not exceeding <math>x<math>, one can prove that for suitable constants <math>c_1<math> and <math>c_2<math>:
<math>c_1\frac x{\log x}prime number theorem.