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A265626 Let S be the set of factorizations of n! where the largest factor is the largest prime <= n, and let f(s) be the least factor in the factorization s. Then a(n) = max f(S).

%I A265626 #8 Jan 01 2016 20:05:48
%S A265626 2,2,2,2,3,4,3,3,4,7,6,7,7,7,7,10,11,11,11,11,11,13,13,13,13,13,13,17,
%T A265626 17,17,17,17,17,17,17,19,19,19,19,23,23,23,23,23,23,29,29,29,29,29,29,
%U A265626 29,29,29,29,29,29,31,31,31,31,31,31,31,31,37,37
%N A265626 Let S be the set of factorizations of n! where the largest factor is the largest prime <= n, and let f(s) be the least factor in the factorization s. Then a(n) = max f(S).
%F A265626 a(n) > 5 for n > 10. a(n) < A007918(ceiling(A007917(n)/2)).
%e A265626 2! = 2
%e A265626 3! = 2 * 3
%e A265626 4! = 2^3 * 3
%e A265626 5! = 2 * 3 * 4 * 5
%e A265626 6! = 3^2 * 4^2 * 5
%e A265626 7! = 4 * 5 * 6^2 * 7
%e A265626 8! = 3 * 4^3 * 5 * 6 * 7
%e A265626 9! = 3 * 4^2 * 5 * 6^3 * 7
%e A265626 10! = 4^2 * 5^2 * 6^4 * 7
%e A265626 11! = 7 * 8^2 * 9^2 * 10^2 * 11
%e A265626 12! = 6^5 * 7 * 8 * 10^2 * 11
%e A265626 13! = 7 * 8^2 * 9^2 * 10^2 * 11 * 12 * 13
%e A265626 14! = 7^2 * 8 * 9 * 10^2 * 11 * 12^3 * 13
%e A265626 15! = 7^2 * 9 * 10^3 * 11 * 12^4 * 13
%e A265626 16! = 7^2 * 10^3 * 11 * 12^6 * 13
%e A265626 17! = 10 * 11 * 12^4 * 13 * 14^2 * 15^2 * 16 * 17
%o A265626 (PARI) f(n,mn,mx)=if(n%mn,return(0)); n/=mn; if(n==1, return(1)); for(k=mn,mx, if(f(n,k,mx), return(1))); 0
%o A265626 a(n)=if(n<6,return(2)); my(p=precprime(n),q=nextprime(p/2),N=n!); forprime(r=q+1,p-1, N/=r^valuation(N,r)); forstep(k=q,3,-1, if(f(N,k,p), return(k)))
%Y A265626 Cf. A001055, A007917, A007918, A076716, A177333.
%K A265626 nonn
%O A265626 2,1
%A A265626 _Charles R Greathouse IV_, Dec 10 2015