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Numbers n such that 1.5^n is closer to an integer than 1.5^m for any 0 < m < n.
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%I #14 Feb 05 2016 17:56:56

%S 1,2,4,29,46,58,95,153,157,163,455,1060,1256,2677,3328,12429,49304,

%T 112896,129638,164000

%N Numbers n such that 1.5^n is closer to an integer than 1.5^m for any 0 < m < n.

%C Zudilin proves that the distance from 1.5^n to the nearest integer is at least 0.5803^n for large enough n; it seems that n > 4 suffices. (The "large enough" constant in the proof is effective but not explicit.)

%H Wadim Zudilin, <a href="http://www.emis.de/journals/JTNB/2007-1/article19.pdf">A new lower bound for ∥(3/2)^k∥</a>, J. Théor. Nombres Bordeaux 19:1 (2007), pp. 311-323.

%e 1.5^29 = 127834.039... which is within 0.039... of an integer, yielding a(4) = 29.

%e 1.5^46 = 125949925.968... which is within 0.031... of an integer, yielding a(5) = 46.

%o (PARI) f(x)=x=frac(x); if(x>1/2,1-x,x)

%o t=r=1;for(n=1,1e6, tt=f(t*=3/2); if(tt<r,r=tt;print1(n", ")))

%Y Cf. A079490, A080052.

%K nonn

%O 1,2

%A _Charles R Greathouse IV_, Jan 11 2016