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Numbers k such that the fractional part of (10/9)^k is less than 1/k.
10

%I #14 Mar 26 2019 19:27:53

%S 1,2,7,62,324,1647,3566,5464,8655,8817,123956,132891,182098,566593,

%T 2189647,2189648,3501843,3501844

%N Numbers k such that the fractional part of (10/9)^k is less than 1/k.

%C Numbers k such that fract((10/9)^k) < 1/k, where fract(x) = x-floor(x).

%C The next such number must be greater than 2*10^5.

%C a(19) > 10^7. - _Robert Price_, Mar 24 2019

%C Given a number k that is not only a term of this sequence but also has the property that the integer part of (10/9)^k is divisible by 9, we can expect that k+1 will likely also be a term of the sequence. E.g., k = 2189647 is a term because fract((10/9)^k) = 0.000000373557... < 0.000000456694... = 1/k, and since floor((10/9)^k) is divisible by 9, the integer and fractional parts of (10/9)^(k+1) will be exactly 10/9 times the integer and fractional parts of (10/9)^k, respectively, yielding a fractional part (10/9) * 0.000000373557... = 0.000000415064... < 0.000000456694... = 1/(k+1), so k+1 = 2189648 is also a term. - _Jon E. Schoenfield_, Mar 24 2019

%e a(3) = 7 since fract((10/9)^7) = 0.09075... < 1/7, but fract((10/9)^k) >= 1/k for 3 <= k <= 6.

%t Select[Range[1000], FractionalPart[(10/9)^#] < (1/#) &] (* _G. C. Greubel_, Aug 24 2016 *)

%Y Cf. A153662, A153670, A153678, A153686, A153698, A154130, A153702, A153710, A153718.

%K nonn,more

%O 1,2

%A _Hieronymus Fischer_, Jan 06 2009

%E a(14)-a(18) from _Robert Price_, Mar 24 2019