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Numbers k such that the fractional part of (1024/1000)^k is less than 1/k.
12

%I #16 Mar 03 2020 06:57:53

%S 1,2,3,4,5,6,585,1164,1707,522271,3675376,3906074,9424094

%N Numbers k such that the fractional part of (1024/1000)^k is less than 1/k.

%C Numbers k such that fract((1024/1000)^k) < 1/k, where fract(x) = x-floor(x).

%C The next such number must be greater than 5*10^5.

%C a(14) > 10^7. - _Robert Price_, Mar 16 2019

%e a(7) = 585 since fract((1024/1000)^585) = 0.00139... < 1/585, but fract((1024/1000)^k) >= 1/k for 7 <= k <= 584.

%t Select[Range[2000], FractionalPart[(1024/1000)^#] < (1/#) &] (* _G. C. Greubel_, Aug 24 2016; corrected by _Robert Price_, Mar 16 2019 *)

%o (PARI) isok(n) = frac((1024/1000)^n) < 1/n \\ _Michel Marcus_, Aug 06 2013

%Y Cf. A153662, A153670, A153682, A154130, A153686, A153694, A153702, A153710, A153718.

%K nonn,more

%O 1,2

%A _Hieronymus Fischer_, Jan 06 2009

%E a(10)-a(13) from _Robert Price_, Mar 16 2019