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Concept
6
Summary
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.
Six is the smallest positive integer which is neither a square number nor a prime number. It is the second smallest composite number after four, equal to the sum and the product of its three proper divisors (, and ). As such, 6 is the only number that is both the sum and product of three consecutive positive numbers. It is the smallest perfect number, which are numbers that are equal to their aliquot sum, or sum of their proper divisors. It is also the largest of the four all-Harshad numbers (1, 2, 4, and 6).
6 is a pronic number and the only semiprime to be. It is the first discrete biprime (2 × 3) which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6n ± 1 for n ≥ 1.
As a perfect number:
6 is related to the Mersenne prime 3, since 2^1(2^2 – 1) = 6. (The next perfect number is 28.)
6 is the only even perfect number that is not the sum of successive odd cubes.
6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, .
Six is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist; sixty (10 × 6) and ninety (15 × 6) are the next two.
All integers that are multiples of 6 are pseudoperfect (all multiples of a pseudoperfect number are pseudoperfect). Six is also the smallest Granville number, or -perfect number.
Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". Six is a congruent number.
6 is the second primary pseudoperfect number, and harmonic divisor number. It is also the second superior highly composite number, and the last to also be a primorial.
There are six different ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).
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