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A065560 a(n) is the smallest integer k such that floor((1+1/n)^(k+1))/floor((1+1/n)^k) = 1+1/n.

%I A065560 #16 Dec 11 2024 11:34:27
%S A065560 2,4,7,9,12,15,18,21,25,28,40,35,39,42,46,50,54,58,62,66,70,74,78,83,
%T A065560 87,91,95,100,104,109,113,118,122,127,131,136,141,145,150,155,159,164,
%U A065560 169,174,179,183,188,193,198,203,208,213,218,223,228,233,238,243,248,253
%N A065560 a(n) is the smallest integer k such that floor((1+1/n)^(k+1))/floor((1+1/n)^k) = 1+1/n.
%C A065560 a(n) is growing roughly like prime(n). a(n) < a(n+1) except for n = 12. (Is this the only exception?)
%C A065560 a(n) < a(n+1) except for n = 12, 108, 266, ... - Boris Gourevitch (boris(AT)pi314.net), Dec 04 2001
%C A065560 Conjecture: a(n)+n > prime(n).
%H A065560 Harry J. Smith, <a href="/A065560/b065560.txt">Table of n, a(n) for n=2..800</a>
%F A065560 Asymptotic (conjectured) formula: a(n)=n*log(n)+o(log(n)).
%e A065560 a(5) = 9 because 9 is the first integer satisfying floor((6/5)^(9+1))/floor((6/5)^9) = 6/5.
%o A065560 (PARI) a(n) = { my(k=1, f=(n + 1)/n); while((floor(f^(k + 1))/floor(f^k)) != f, k++); k } \\ _Harry J. Smith_, Oct 22 2009
%Y A065560 Cf. A065554, A065564.
%K A065560 nonn
%O A065560 2,1
%A A065560 _Benoit Cloitre_, Nov 29 2001
%E A065560 Terms a(53) - a(61) from _Harry J. Smith_, Oct 22 2009