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%I #8 Mar 18 2019 21:50:26
%S 2,3,4,5,6,9,10,11,17,19,60,66,89,576,3069,3901,6270,7542,13046,13215,
%T 27952,43110,46605,55413,61421,93159,96004,164035,248150,298207,
%U 301985,749378,1149838,1948414,1959239,2045876,2174103,2819116,3666855,3935292,4650787,4746097
%N Numbers n>1 such that the difference between log(n) and its best rational approximation as x/y with y<=n produces a new minimum of abs(log(n)-x/y). x/y is provided as A306976/A306977.
%e k L=log(k) x/y |L - x/y|
%e 2 0.6931471... 1/2 0.1931471... new minimum
%e 3 1.0986122... 1/1 0.0986122... new minimum
%e 4 1.3862943... 4/3 0.0529610... new minimum
%e 5 1.6094379... 8/5 0.0094379... new minimum
%e 6 1.7917594... 9/5 0.0082405... new minimum
%e 7 1.9459101... 2/1 0.0540898...
%e 8 2.0794415... 17/8 0.0455584...
%e 9 2.1972245... 11/5 0.0027754... new minimum
%e 10 2.3025850... 23/10 0.0025850... new minimum
%e 11 2.3978952... 12/5 0.0021047... new minimum
%e 12 2.4849066... 5/2 0.0150933...
%e 13 2.5649493... 18/7 0.0064792...
%e 14 2.6390573... 29/11 0.0026936...
%e 15 2.7080502... 19/7 0.0062355...
%e 16 2.7725887... 36/13 0.0033579...
%e 17 2.8332133... 17/6 0.0001199... new minimum
%e .
%e a(1..9) = [2, 3, 4, 5, 6, 9, 10, 11, 17],
%e A306976(1..9) = [1, 1, 4, 8, 9, 11, 23, 12, 17],
%e A306977(1..9) = [2, 1, 3, 5, 5, 5, 10, 5, 6].
%o (PARI) dmin=1; for(k=2,5000000,L=log(k);d=abs(L-bestappr(L,k));if(d<dmin,dmin=d;print1(k,", ")))
%Y Cf. A306972, A306976, A306977.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Mar 18 2019