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 with n > 1 and primes is practical if and only if p₁ = 2 and for 1 (one) is the natural number following 0 and preceding 2. ... 2 (two) is the natural number following 1 and preceding 3. ... 4 (four) is the natural number following 3 and preceding 5. ... 6 (six) is the natural number following 5 and preceding 7. ... 8 (eight) is the natural number following 7 and preceding 9. ... 12 (twelve) is the natural number following 11 and preceding 13. ... 16 (sixteen) is the natural number following 15 and preceding 17. ... 18 (eighteen) is the natural number following 17 and preceding 19. ... 20 (twenty) is the natural number following 19 and preceding 21. ... 24 (twenty-four) is the natural number following 23 and preceding 25. ... 28 (twenty-eight) is the natural number following 27 and preceding 29. ... 30 (thirty) is the natural number following 29 and preceding 31. ... 32 is the natural number following 31 and preceding 33. ... 36 is the natural number following 35 and preceding 37. ... 40 is the natural number following 39 and preceding 41. ... 42 is the natural number following 41 and followed by 43. ... 48 is the natural number following 47 and preceding 49. ... 52 is the natural number following 51 and preceding 53. ... Cardinal fifty-four Ordinal 54th (fifty-fourth) Factorization Divisors 2, 3, 6, 9, 18, 27 Roman numeral LIV Binary 110110 Hexadecimal 36 Fifty-four (54) is the natural number following 53 and preceding 55. ...

For example, any even perfect numbers is also a practical number. In mathematics, a perfect number is an integer which is the sum of its proper positive divisors, excluding itself. ...

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: In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ...

a formula which resembles the prime number theorem. In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...