4

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+ If we add 4 and the 4th Fibonacci number (which is prime), we find the 4th Lucas number (which is the 4th prime number). [Capelle]

+ If S(n) is the sum of the first n primes, then the limit of S(2n)/S(n) = 4, as n approaches infinity. [Capelle]

+ 4 is the smallest number n such that n and n! are product of factorials of primes (4 = 2!2! and 4! = 2!2!3!). [Capelle]

+ There are 4 known positive integers n such that the sum of all primes smaller or equal to n divides n(n+1)/2. This sum also divides the sum of all nonprimes smaller or equal to n. Note that 4 is the smallest positive integer with this property. [Capelle]

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