Geodesic
A natural prime-generating recurrence
Journal of integer sequences, Tome 11 (2008) no. 2
For the sequence defined by $a(n) = a(n-1) + gcd(n,a(n-1))$ with $a(1) = 7$ we prove that $a(n) - a(n-1)$ takes on only 1's and primes, making this recurrence a rare "naturally occurring" generator of primes. Toward a generalization of this result to an arbitrary initial condition, we also study the limiting behavior of $a(n)/n$ and a transience property of the evolution.
Classification : 11A41, 11B37
Keywords: prime-generating recurrence, prime formulas, discrete dynamical systems, greatest common divisor 
@article{JIS_2008__11_2_a0,
     author = {Rowland,  Eric S.},
     title = {A natural prime-generating recurrence},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {2},
     zbl = {1204.11015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_2_a0/}
}
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Rowland,  Eric S. A natural prime-generating recurrence. Journal of integer sequences, Tome 11 (2008) no. 2. http://geodesic.mathdoc.fr/item/JIS_2008__11_2_a0/