Hmm, interesting — I’d heard of 22 and 58, but not of 6, 25, 37.

It was especially the famous case of 163 that got me thinking: If the sequence of frac(Q(n)), n=1,2,3, is uniformly distributed in [0,1] then the sequence of recordholders n_k must be infinite.

Indeed. A weaker assumption is that frac(Q(n)) is dense in [0,1]; this may be amenable to ergodic theory. Sincerely, Adam P. Goucher

—Dan

On Mar 1, 2014, at 7:40 PM, Fred W. Helenius <fredh@ix.netcom.com> wrote:

...

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

...

Question:

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

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.

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