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Numbers k such that the fractional part of (101/100)^k is less than 1/k.
14

%I #15 Dec 24 2021 08:17:02

%S 1,2,3,4,5,6,7,8,9,70,209,241,378,2697,4806,173389,529938,1334508,

%T 1572706,7840546,15896994,20204295,71074288,119325567

%N Numbers k such that the fractional part of (101/100)^k is less than 1/k.

%C Numbers k such that fract((101/100)^k) < 1/k, where fract(x) = x-floor(x).

%C The next term is greater than 2*10^8.

%e a(10) = 70 since fract((101/100)^70) = 0.006... < 1/10, but fract((101/100)^k) > 0.1 >= 1/k for 10 <= k <= 69.

%t Select[Range[1000], FractionalPart[(101/100)^#] < (1/#) &] (* _G. C. Greubel_, Aug 24 2016 *)

%o (Python)

%o from itertools import count, islice

%o def A153670gen(): # generator of terms

%o k10, k11 = 100, 101

%o for k in count(1):

%o if (k11 % k10)*k < k10:

%o yield k

%o k10 *= 100

%o k11 *= 101

%o A153670_list = list(islice(A153670gen(),16)) # _Chai Wah Wu_, Dec 23 2021

%Y Cf. A153662, A154130, A153674, A153678, A153686, A153694, A153702, A153710, A153718.

%K nonn,more

%O 1,2

%A _Hieronymus Fischer_, Jan 06 2009

%E a(18)-a(24) from _Robert Gerbicz_, Nov 29 2010