Re: [math-fun] Question about uniform distribution (mod 1) of a sequence

On 3/1/2014 4:02 PM, Dan Asimov wrote:

Question:

Let Q(n) := exp(pi*sqrt(n)) for n = 1,2,3,Â….

What can be said about the distribution of the sequence frac(Q(n)) = Q(n) (mod 1) ???

Since my next birthday, #n, satisfies exp(pi*sqrt(n)) is close to an integer, I was wondering what the sequence (letting f(x) := min{frac(x),1-frac(x)} be the distance of x to its nearest integer):

n_1 = 1, and

n_(k+1) := min{n | f(Q(n))) < f(Q(n_k))}

looks like. I.e., n for which f(Q(x)) is a record low.

The sequence of integers yielding record lows is oeis.org/A069014 . Apparently no further value is known that beats 163, so the sequence shows only nine terms: 1, 2, 6, 17, 22, 25, 37, 58, 163. -- Fred W. Helenius fredh@ix.netcom.com