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(6)=4, since 1-(1/5)=0.8>fract((11/10)^A153687(6))=fract((11/10)^6)=0.771...>=1-(1/4).
a(30)=82, since fract((1024/1000)^82)= 0.99191990..., but fract((1024/1000)^k)<0.9893 for 1<=k<=81; thus fract((1024/1000)^82)>fract((1024/1000)^k) for 1<=k<82 and 82 is the minimal exponent >29 with this property.
$MaxExtraPrecision = 10000; p = 0; Select[Range[1, 50000], If[FractionalPart[(1024/1000)^#] > p, p = FractionalPart[(1024/1000)^#]; True] &] (* Robert Price, Mar 16 2019 *)
A153679_list, m, n, k, q = [], 1, 1024, 1000, 0 while m < 10**4: r = n % k if r > q: q = r A153679_list.append(m) m += 1 n *= 1024 k *= 1000 q *= 1000 # Chai Wah Wu, May 16 2020
a(7)=13, since fract((10/9)^13) = 0.93..., but fract((10/9)^k) < 0.89 for 1 <= k <= 12; thus fract((10/9)^13) > fract((10/9)^k) for 1 <= k < 13 and 13 is the minimal exponent > 6 with this property.
$MaxExtraPrecision = 100000; p = 0; Select[Range[1, 20000], If[FractionalPart[(10/9)^#] > p, p = FractionalPart[(10/9)^#]; True] &] (* Robert Price, Mar 24 2019 *)
A153695_list, m, m10, m9, q = [], 1, 10, 9, 0 while m < 10**4: r = m10 % m9 if r > q: q = r A153695_list.append(m) m += 1 m10 *= 10 m9 *= 9 q *= 9 # Chai Wah Wu, May 16 2020
a(2)=8, since fract(e^8)= 0.9579870417..., but fract(e^k)<=0.7182818... for 1<=k<=7; thus fract(e^8)>fract(e^k) for 1<=k<8 and 8 is the minimal exponent > 1 with this property. [_Hieronymus Fischer_, Jan 06 2009]
a = 0; Do[b = N[ FractionalPart[ N[ E, 2^12]^n], 24]; If[b > a, Print[n]; a = b], {n, 1, 9400}] (* Robert G. Wilson v, Mar 16 2004 *)
E=exp(1); /* use sufficient precision! */
ym=0;for(i=1,1000,x=E^i;y=x-floor(x);if(y>ym,print1(","i);ym=y))
a(2)=23, since 1-(1/24) = 0.9583...> fract(e^A091560(2)) = fract(e^8) = 0.95798.. >= 0.95652... >= 1-(1/23).
$MaxExtraPrecision = 100000; A091560 = {1,8,19,76,166,178,209,1907,20926,22925,32653,119136}; Floor[1/(1-FractionalPart[E^A091560])] (* Robert Price, Apr 18 2019 *)
a(3)=15, since fract(Pi^15)= 0.9693879984..., but fract(Pi^k)<=0.8696... for 1<=k<=14; thus fract(Pi^15)>fract(Pi^k) for 1<=k<15 and 15 is the minimal exponent > 2 with this property.
$MaxExtraPrecision = 100000; p = 0; Select[Range[1, 10000], If[FractionalPart[Pi^#] > p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 25 2019 *)
a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153711(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).
$MaxExtraPrecision = 100000; A153711 = {1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710}; Floor[1/(1-FractionalPart[Pi^A153711])] (* Robert Price, Apr 18 2019 *)
a(5) = 16, since 1-(1/17) = 0.941176... > fract((Pi-2)^A153719(5)) = fract((Pi-2)^5) = 0.9389... >= 0.9375 = 1-(1/16).
$MaxExtraPrecision = 100000; A153719 = {1, 2, 3, 4, 5, 39, 56, 85, 557, 911, 2919, 2921, 4491, 11543, 15724, 98040, 110932, 126659}; Floor[1/(1 - FractionalPart[(Pi - 2)^A153719])] (* Robert Price, Apr 18 2019 *)
a(6)=39, since fract((Pi-2)^39)= 0.9586616565..., but fract((Pi-2)^k)<=0.9389018... for 1<=k<=38; thus fract((Pi-2)^39)>fract((Pi-2)^k) for 1<=k<39 and 39 is the minimal exponent > 5 with this property.
$MaxExtraPrecision = 100000; p = 0; Select[Range[1, 10000], If[FractionalPart[(Pi - 2)^#] > p, p = FractionalPart[(Pi - 2)^#]; True] &] (* Robert Price, Mar 26 2019 *)
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