cp's OEIS Frontend

A153694 Numbers k such that the fractional part of (10/9)^k is less than 1/k.

%I A153694 #14 Mar 26 2019 19:27:53
%S A153694 1,2,7,62,324,1647,3566,5464,8655,8817,123956,132891,182098,566593,
%T A153694 2189647,2189648,3501843,3501844
%N A153694 Numbers k such that the fractional part of (10/9)^k is less than 1/k.
%C A153694 Numbers k such that fract((10/9)^k) < 1/k, where fract(x) = x-floor(x).
%C A153694 The next such number must be greater than 2*10^5.
%C A153694 a(19) > 10^7. - _Robert Price_, Mar 24 2019
%C A153694 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 A153694 a(3) = 7 since fract((10/9)^7) = 0.09075... < 1/7, but fract((10/9)^k) >= 1/k for 3 <= k <= 6.
%t A153694 Select[Range[1000], FractionalPart[(10/9)^#] < (1/#) &] (* _G. C. Greubel_, Aug 24 2016 *)
%Y A153694 Cf. A153662, A153670, A153678, A153686, A153698, A154130, A153702, A153710, A153718.
%K A153694 nonn,more
%O A153694 1,2
%A A153694 _Hieronymus Fischer_, Jan 06 2009
%E A153694 a(14)-a(18) from _Robert Price_, Mar 24 2019