This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(3) = 15, since fract(Pi^15) = 0.969... > 0.933... = 1 - (1/15), but fract(Pi^k) <= 1 - (1/k) for 3 <= k <= 14.
Select[Range[1000], N[FractionalPart[Pi^#], 100] > 1 - (1/#) &] (* G. C. Greubel, Aug 25 2016 *)
a(2) = 23, since 1-(1/24) = 0.9583... > fract(e^A153704(2)) = fract(e^8) = 0.95798... >= 0.95652... >= 1-(1/23).
A153704 = {1, 8, 19, 178, 209, 1907, 32653, 119136, 220010}; Table[fp = FractionalPart[E^A153704[[n]]]; m = Floor[1/fp]; While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153704]}] (* Robert Price, May 10 2019 *)
a(2)=93, since 1-(1/94)=0.98936...>fract((1024/1000)^A153680(2))=fract((1024/1000)^29)=0.98929...>=1-(1/93).
a(3)=15, since 1-(1/16)=0.9375>fract((10/9)^A153696(3))=fract((10/9)^13)=0.9341...>=1-(1/15).
a(2)=19, since 1-(1/20)=0.95>fract((11/10)^A153688(2))=fract((11/10)^7)=0.9487...>=1-(1/19).
a(2)=76, since 1-(1/77)=0.9870...>fract((101/100)^A153672(2))=fract((101/100)^69)=0.98689...>=1-(1/76).
a(3)=88, since 1-(1/89)=0.988764...>fract((4/3)^A154133(3))=fract((4/3)^8)=0.988721...>0.988636...=1-(1/88).
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