Contributions to Online Encyclopedia of Integer Sequences

Four consecutive primes of the form (6k-1)

Four  consecutive primes of the form (6k+1)

Denominators of Zeta(2n)/π^(2n)

Solution to a^5 + b^5 = c^5 + d^5

Integers k that makes 6k+1 prime

a(n) = (1/e) * sum_{j>1} j!/(j-n)!^2

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.

 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*(n+2) )^2. Added:  a(n) = (A005563(n))^2 

 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).