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%I #8 Mar 18 2019 08:11:47
%S 2,3,5,8,10,15,17,19,20,21,28,44,68,72,90,107,114,115,156,159,164,267,
%T 278,294,336,427,455,464,592,644,665,754,998,1040,1139,1173,1278,1314,
%U 1335,1463,1503,1593,1943,2175,2262,2520,2642,2873,2978,3003,3047,3181,3197
%N Nonsquare numbers n such that the difference between sqrt(n) and its best rational approximation as x/y with y<=n produces a new minimum of abs(sqrt(n)-x/y). x/y is provided as A306973/A306974.
%e k s=sqrt(k) x/y |s - x/y|
%e 2 1.4142135... 3/2 0.0857864... new minimum
%e 3 1.7320508... 5/3 0.0653841... new minimum
%e 5 2.2360679... 9/4 0.0139320... new minimum
%e 6 2.4494897... 5/2 0.0505102...
%e 7 2.6457513... 8/3 0.0209153...
%e 8 2.8284271... 17/6 0.0049062... new minimum
%e 10 3.1622776... 19/6 0.0043890... new minimum
%e 11 3.3166247... 10/3 0.0167085...
%e 12 3.4641016... 7/2 0.0358983...
%e 13 3.6055512... 18/5 0.0055512...
%e 14 3.7416573... 15/4 0.0083426...
%e 15 3.8729833... 31/8 0.0020166... new minimum
%e 17 4.1231056... 33/8 0.0018943... new minimum
%e 18 4.2426406... 17/4 0.0073593...
%e 19 4.3588989... 61/14 0.0017560... new minimum
%e 20 4.4721359... 76/17 0.0015477... new minimum
%e .
%e a(1)..a(9) = [2, 3, 5, 8, 10, 15, 17, 19, 20]
%e A306973(1..9) = [3, 5, 9, 17, 19, 31, 33, 61, 76]
%e A306974(1..9) = [2, 3, 4, 6, 6, 8, 8, 14, 17]
%o (PARI) dmin=1; for(k=2,3200,if(!issquare(k),s=sqrt(k);d=abs(s-bestappr(s,k));if(d<dmin,dmin=d;print1(k,", "))))
%Y Cf. A306973, A306974.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Mar 18 2019
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