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).

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, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37

Offset: 2

Author

Charles R Greathouse IV, Dec 10 2015

Keywords

• nonn

Examples

			2! = 2
3! = 2 * 3
4! = 2^3 * 3
5! = 2 * 3 * 4 * 5
6! = 3^2 * 4^2 * 5
7! = 4 * 5 * 6^2 * 7
8! = 3 * 4^3 * 5 * 6 * 7
9! = 3 * 4^2 * 5 * 6^3 * 7
10! = 4^2 * 5^2 * 6^4 * 7
11! = 7 * 8^2 * 9^2 * 10^2 * 11
12! = 6^5 * 7 * 8 * 10^2 * 11
13! = 7 * 8^2 * 9^2 * 10^2 * 11 * 12 * 13
14! = 7^2 * 8 * 9 * 10^2 * 11 * 12^3 * 13
15! = 7^2 * 9 * 10^3 * 11 * 12^4 * 13
16! = 7^2 * 10^3 * 11 * 12^6 * 13
17! = 10 * 11 * 12^4 * 13 * 14^2 * 15^2 * 16 * 17
		

Crossrefs

Cf. A001055, A007917, A007918, A076716, A177333.

Programs

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
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)))

Formula

a(n) > 5 for n > 10. a(n) < A007918(ceiling(A007917(n)/2)).