A145355 - OEIS

%I #9 May 13 2013 01:54:10

%S 1,1,5,11,3,71,2,1,8,20,5,1,2,5,1,2,2,1,1,3,1,1,8,2,13,22,1,1,3,2,2,3,

%T 2,2,1,2,3,2,3,1,9,2,2,1,2,1,1,2,2,1,6,1,1,4,2,2,2,3,21,2,1,1,1,1,2,2,

%U 6,8,4,7,1,2,2,1,3,1,1,9,2,1,2,4,3,5,1,1,2,5,13,6

%N a(n) = round(round(sqrt(n!)/abs(round(sqrt(n!))^2 - n!))).

%C This sequence suggests that the distance between a factorial and the closest power is tightly bounded. Generated by _Ed Pegg Jr_ in response to three Alexander R. Povolotsky conjectures: 1)n! + n^2 != m^2 (except for trivial case with n=0, m=1) per conducted calculations doesn't yield any solutions from n=1 to n= 200,000 2)n! + Sum(j^2, j=1, j=n) != m^2 per conducted calculations doesn't yield any solutions from n=1 to n= 2,000,000 3)n! + prime(n) != m^k is too difficult to cover by exhaustive calculations ...

%H Charles R Greathouse IV, <a href="/A145355/b145355.txt">Table of n, a(n) for n = 2..10000</a>

%o (PARI) a(n)=my(s=round(sqrt(n!)));s\/abs(s^2-n!) \\ _Charles R Greathouse IV_, Dec 20 2011

%K nonn

%O 2,3

%A _Alexander R. Povolotsky_, Oct 09 2008