OFFSET
1,2
COMMENTS
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.)
LINKS
Wadim Zudilin, A new lower bound for ∥(3/2)^k∥, J. Théor. Nombres Bordeaux 19:1 (2007), pp. 311-323.
EXAMPLE
1.5^29 = 127834.039... which is within 0.039... of an integer, yielding a(4) = 29.
1.5^46 = 125949925.968... which is within 0.031... of an integer, yielding a(5) = 46.
PROG
(PARI) f(x)=x=frac(x); if(x>1/2, 1-x, x)
t=r=1; for(n=1, 1e6, tt=f(t*=3/2); if(tt<r, r=tt; print1(n", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Charles R Greathouse IV, Jan 11 2016
STATUS
approved