Gaps between prime-roots

[Neil Fernandez]

In message <007501c2ca4d$daa5d9b0$7b01a8c0 at pcnt>, Rainer Rosenthal
<r.rosenthal at web.de> writes
>In the course of thread "Primes..." in sci.math I proposed
>the following:
>
>         1, 113, 1327, ... ???  
>
>where a(n) = smallest natural such that
>sqrt(P_(k+1)) - sqrt(P_k) < 1/n  for all k with P_k > a(n)

Are the terms 113 and 1327 proven?

Another sequence can be got if a(n) is just the first prime for which
sqrt(p(n+1))-sqrt(p(n))<1/n, namely

2,2,2,17,29,41,59,71,101,101,137,149,179,197,227,269,...

(numbers appearing more than once:
101,419,809,1019,1229,1607,2549,2969,6089,15137)

(numbers appearing more than twice: 809,...)

Generalising the index of the root to 1/t:
a(n) is the first prime for which (p(n+1))^(1/t) - (p(n))^(1/t) <1/n,

t=3: 2,2,2,2,2,11,11,17,17,17,29,29,29,29,41,41,41,41,...

t=1.5: 2,2,71,179,311,521,821,1229,1787,2381,3167,4127,5231,...

t=1.25: 2,347,2549,10859,...

What is the smallest t such that at least one number >2 appears in the
sequence more than once?

t=1.808: 809,...

Neil

-- 
Neil Fernandez