A naturally occurring prime generator

Pyth, 14 13 bytes

meaYhP+sY-5dQ

Uses a(n) = Lpf(6 - n + sum(a(i) for i in range(1, n)) where Lpf means least prime factor.

Try it here online.

Python 3.5.0b1+, 95 93 bytes

Link to Python 3.5.0b1+ release

import math
def f(k,n=2,a=7,L=[]):x=math.gcd(n,a);return k and f(k-1%x,n+1,a+x,L+1%x*[x])or L

A direct implementation of the recurrence, featuring:

• Our good friend 1%x, and
• math.gcd, as opposed to fractions.gcd.

Julia, 110 bytes

n->(a(n)=(n≥1&&(n==1?7:a(n-1)+gcd(n,a(n-1))));i=2;j=0;while j<n x=a(i)-a(i-1);x>1&&(j+=1;println(x));i+=1end)

Ungolfed:

function a(n::Int)
    n ≥ 1 && (n == 1 ? 7 : a(n-1) + gcd(n, a(n-1)))
end

function f(n::Int)
    i = 2;
    j = 0;
    while j < n
        x = a(i) - a(i-1)
        if x > 1
            j += 1
            println(x)
        end
        i += 1
    end
end