Prime-Generating Recurrence
Prime-Generating Recurrence
This Demonstration explores solutions of the recurrence through the difference sequence , which exhibits complex behavior. For the initial condition , the sequence consists entirely of s and primes, making this recurrence a rare "naturally occurring" generator of primes.
a(n)=a(n-1)+gcd(n,a(n-1))
a(n)-a(n-1)
a(1)=7
a(n)-a(n-1)
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This result is not true in general: for example, letting produces , and letting produces . However, for these initial conditions, the difference sequence eventually consists entirely of s and primes. It is an unsolved problem to determine whether all initial conditions eventually produce only s and primes.
a(1)=532
a(18)-a(17)9
a(1)=800
a(21)-a(20)21
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You can choose to view all terms of the difference sequence or only the terms which are not .
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