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Practical 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 $n=p_1^{\alpha_1}...p_k^{\alpha_k}$ with n > 1 and $p_1 primes is practical if and only if p1 = 2 and for $i=2,\dots,k$

$p_i\leq\sigma(p_1^{\alpha_1}\dots p_{i-1}^{\alpha_{i-1}})+1$

For example, any even perfect 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 p(x) is the enumerating function of practical numbers, i.e., the number of practical numbers not exceeding x, one can prove that for suitable constants c1 and c2:

$c_1\frac x{\log x}

a formula which resembles the prime number theorem.

03-10-2013 05:06:04