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One of the most pointless forum games ever to be created. But meh, whatever. Just keep counting. Next post should be 2, then 3, then 4, etc.

 

1

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Six is the second smallest composite number, its proper divisors being 1, 2 and 3.

 

Since six equals the sum of these proper divisors, six is the smallest perfect number, Granville number, and \mathcal{S}-perfect number.[1][2] As a perfect number, 6 is related to the Mersenne prime 3, since 21(22 - 1) = 6. (The next perfect number is 28.) It is the only even perfect number that is not the sum of successive odd cubes.[3] Being perfect, six is the root of the 6-aliquot tree, and is itself the aliquot sum of only one number; the square number, 25. Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler."[4] Six is a congruent number.

 

Six is the first discrete biprime (2.3) and the first member of the (2.q) discrete biprime family.

 

Six is the only number that is both the sum and the product of three consecutive positive numbers.[5]

 

Six is a unitary perfect number, a harmonic divisor number and a highly composite number. The next highly composite number is 12.

 

5 and 6 form a Ruth-Aaron pair under either definition.

 

The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.

 

S6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in 1-1 correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6.

 

6 similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

A cube has 6 faces

 

6 is the largest of the four all-Harshad numbers.

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Fifteen is a triangular number, a hexagonal number, a pentatope number and the 4th Bell number. Fifteen is the double factorial of 5. It is a composite number; its proper divisors being 1, 3 and 5. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet (11, 13, 17, 19). 15 is also the number of supersingular primes.

 

15 is the 4th discrete semiprime (3.5) and the first member of the (3.q) discrete semiprime family. It is thus the first odd discrete semiprime. The number proceeding 15; 14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21.

 

The aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members (15,9,4,3,1,0). 15 is the fourth composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree. Fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33.

 

15 and 16 form a Ruth-Aaron pair under the second definition in which repeated prime factors are counted as often as they occur.

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