This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A080189 #4 Mar 30 2012 17:27:40
%S A080189 0,1,2,3,2,4,2,3,5,3,3,4,4,6,3,4,3,4,6,2,5,5,7,4,3,3,5,2,5,5,7,7,6,6,
%T A080189 3,4,6,8,7,5,3,5,2,4,3,6,6,6,4,4,6,3,8,8,8,5,3,7,7,4,4,5,6,5,7,9,8,6,
%U A080189 4,5,4,6,5,4,3,5,4,7,4,7,4,7,2,6,3,7,5,5,3,7,4,9,9,8,9,8,9,4,4,8,8,5,5,4,3
%N A080189 a(n) = k such that f^k(prime(n)) = 2, where f is the mapping of primes > 2 to primes defined by A052248.
%C A080189 Since the largest of all prime factors of the numbers between prime p and the next prime is smaller than p, we have p > f(p) > f^2(p) > ... > 2, so a(n) is finite for all n.
%e A080189 prime(6) = 13, f(13) = 7, f(7) = 5, f(5) = 3, f(3) = 2, so f^4(13) = 2 and a(6) = 4.
%o A080189 (PARI) {forprime(k=2,580,c=0; p=k; while(p>2,q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m<a,m=a)); p=m; c++); print1(c,","))}
%Y A080189 Cf. A052248.
%K A080189 nonn
%O A080189 1,3
%A A080189 _Klaus Brockhaus_, Feb 10 2003