%N A000001 Smallest number whose reciprocal fits in the square-root gap of consecutive primes %C A000001 A difficult open problem is whether the limit of sqrt(P_(n+1)) - sqrt(P_n) is 0. (Jim Ferry in sci.math 30-th of January 2003) %D A000001 Message-ID: <HZc_9.16064$Vf3.firstname.lastname@example.org> Jim Ferry in sci.math %F A000001 a(n) = ceiling(1/(w'-w)) where w=sqrt(p(n)) and w'=sqrt(p(n+1)) %e A000001 a(3) = 3 because p(3)=5, p(4)=7, w=sqrt(5) w'=sqrt(7) and 1/(w'-w)=2.44. %Y A000001 A000040 %O A000001 1 %K A000001 ,easy,nice,nonn, %A A000001 Rainer Rosenthal (email@example.com), Jan 30 2003
The next interesting sequence is:
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)
I don't have a CAS and would be glad to see someone submit this sequence with more and better consolidated entries.