a(n) = Primorial(n) / Product_{k prime<n} k

a(n) = n * Sum_{k prime<=n} k

Number of different ways that a number between two members of a twin prime pair can be expressed as a sum of two smaller such numbers.

Numbers that can be expressed in more than one way as

6xy + x + y with x >= y > 0

a(n) = ((p-1)!/p) - ((p-1)*(p-1)!/p!),

where p is the n-th prime.

From Goldbach's conjecture: a(n) is the number of decompositions of 6n into a sum of two primes.

Integers that are not multiples of 6 and are the sum of two consecutive primes

Multiples of 6 that are not the sum of two consecutive primes.

Pentagonal numbers. a(n) = binomial(n,2) + n^2

Generalized pentagonal numbers. a(n) = Sum_{k=1..n} k/gcd(k,2)

Primes of the form 6k-1. For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y

a(n) = Product_{k=1..n} k^(2k - 1 - n). Added formula: a(n) = Product_{i=1..n} Product_{j=1..i} (i/j)

Numbers k such that 6*k + 1 is prime. Added: For all elements of this sequence there are no (x,y) positive integers such that a(n)=6*x*y+x+y or a(n)=6*x*y-x-y

Numbers k such that 6*k-1 and 6*k+1 are twin composites. Added: All terms can be expressed as (6ab+a+b OR 6cd-c-d) AND (6xy+x-y) for a,b,c,d,x,y positive integers. Example: 20=6*2*2-2-2 AND 20=6*3*1+3-1.

Number of entries in the second cycles of all permutations of {1,2,...,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements. Added: a(n) = n! * Sum_{i=1..n} (Sum_{j=1..i} (j/i))

a(n) = (n-1)! * Sum_{k=1..n} k^k/k!. Added: a(n) = (n-1)!*Sum_{i=1..n} Product_{j=1..i} i/j

Coefficients in g.f. for certain marked mesh patterns. Added:

a(n) = (n-3)! * Sum_{i=1..n-2} (Sum_{j=1..i} (i/j)) AND a(n) = (1/4) * (n-1)! * (2*harmonic(n-1)-1).

Structured tetragonal anti-prism numbers. Added: a(n) = binomial(n,3) + n^3

Suprafactorials: Product of first n hyperfactorials divided by the product of the first n superfactorials. Added: a(n) = Product_{i=1..n} (Product_{j=1..i} binomial(i,j))

a(n) = n!*Sum_{i=1..n} (Sum_{j=1..i} (i/j))

a(n) = (n-1)!*(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)*i/j)

a(n) = gcd(2^n + n!, 3^n + n!, n+1) PRIMALITY TEST.

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