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Let f(x,y) = floor((1+1/x)^y); a(n) is the smallest integer such that f(n,a(n)+1)/f(n,a(n)) = 1+1/n.
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%I #11 Jul 10 2015 00:39:22

%S 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 87,91,95,100,104,109,113,118,122,127,131,136,141,145,150,155,159,164,

%U 169,174,179,183,188,193,198,203,208,213,218,223,228,233,238,243,248,253

%N Let f(x,y) = floor((1+1/x)^y); a(n) is the smallest integer such that f(n,a(n)+1)/f(n,a(n)) = 1+1/n.

%C a(n) is growing roughly like prime(n). a(n) < a(n+1) except for n = 12. (Is this the only exception?)

%C a(n) < a(n+1) except for n = 12, 108, 266, ... - Boris Gourevitch (boris(AT)pi314.net), Dec 04 2001

%C Conjecture: a(n)+n > prime(n).

%H Harry J. Smith, <a href="/A065560/b065560.txt">Table of n, a(n) for n=2..800</a>

%F Asymptotic (conjectured) formula: a(n)=n*log(n)+o(log(n)).

%e a(5) = 9 because 9 is the first integer satisfying floor((6/5)^(9+1))/floor((6/5)^9) = 6/5.

%o (PARI) { for (n=2, 1000, k=1; f=(n + 1)/n; while((floor(f^(k + 1))/floor(f^k)) != f, k++); write("b065560.txt", n, " ", k) ) } \\ _Harry J. Smith_, Oct 22 2009

%Y Cf. A065554, A065564.

%K nonn

%O 2,1

%A _Benoit Cloitre_, Nov 29 2001

%E Terms a(53) - a(61) from _Harry J. Smith_, Oct 22 2009