1,2

Number of primes in (n*(n*log(n))^(1/2)..(n+1)*((n+1)*log(n+1))^(1/2)] semi-open intervals, n >= 1.

The semi-open intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.

a(n) = pi((n+1)^(3/2)*(log(n+1))^(1/2)) - pi(n^(3/2)*(log(n))^(1/2)) since the intervals are semi-open properly.

The n_th interval length is: ~ (1/2)*(n+1/2)^(1/2)*[3*(log(n+1/2))^(1/2)+(log(n+1/2))^(-1/2)] ~ (3/2)*n^(1/2)*(log(n))^(1/2) as n goes to infinity

The n_th interval prime density is: ~ 2/[3*log(n+1/2)+log(log(n+1/2))] ~ 2/(3*log(n)) as n goes to infinity

The expected number of primes for n_th interval is: ~ (n+1/2)^(1/2)*[3*(log(n+1/2))^(1/2)+(log(n+1/2))^(-1/2)]/ [3*log(n+1/2)+log(log(n+1/2))] ~ n^(1/2)/(log(n))^(1/2) as n goes to infinity

Using Excel 2003, for n in [1..1123], I obtain a(n) >= 1 (at least one prime per interval).

CAUTION: I will submit the b-file, but since Excel 2003 is limited to 15 digits precision, the rounding might assign to the wrong interval a prime which is extremely close to the limit of 2 successive intervals. The b-file NEEDS TO BE VERIFIED using interval arithmetic! (SEE NEXT)

CAUTION (ADDENDA): for n in [1..1123], the minimum ratio of... ABS[n^(3/2)*(log(n))^(1/2)-ROUND[n^(3/2)*(log(n))^(1/2)]]/ [n^(3/2)*(log(n))^(1/2)] that I got is 5.04999E-09 which is well above 1E-15 (15 digits limit of Excel 2003), so every interval did not end too close to an integral value and every prime has then been assigned to its proper interval. My b-file should then be reliable.

If it can be proved that each interval always contains at least one prime, this would constitute shorter intervals than A143898(n) as n gets large.

The sequence A166363 gives even shorter intervals that seem to always contain at least one prime (for n > 1)!

Daniel Forgues, Table of n, a(n) for n=1..1123

Cf. A143898, A134034, A143935 (for primes between successive n^K, for different K.)

Cf. A144140 (showing that for n^K, K=3/2, some intervals fails to contain primes.)

Cf. A166363 (for primes in even shorter intervals.)

Cf. A014085 (for primes between succesive squares.)

Cf. A000720.

Sequence in context: A001468 A014675 A107362 this_sequence A022303 A113189 A143098

Adjacent sequences: A166329 A166330 A166331 this_sequence A166333 A166334 A166335

nonn

Daniel Forgues (squid(AT)zensearch.com), Oct 12 2009

Corrected and edited by Daniel Forgues (squid(AT)zensearch.com), Oct 14 2009

Edited by Daniel Forgues (squid(AT)zensearch.com), Oct 20 2009