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Number of primes between prime(n) and prime(n) + sqrt(prime(n)), where prime(n) is the n-th prime.
3

%I #13 Mar 13 2023 17:54:17

%S 1,0,1,0,1,0,1,1,0,1,0,2,2,1,1,1,1,1,2,2,1,1,1,1,2,3,3,2,1,0,2,2,1,1,

%T 2,2,2,2,2,2,2,2,3,2,2,1,1,3,4,3,2,2,1,2,3,3,4,3,3,2,1,1,3,2,1,1,3,3,

%U 3,3,2,2,3,3,3,3,2,2,3,2,4,3,4,3,3,4,4,3,3,2,2,3,4,3,3,3,2,2,1,3

%N Number of primes between prime(n) and prime(n) + sqrt(prime(n)), where prime(n) is the n-th prime.

%C Conjecture: if prime(n)>=127, there is always at least one prime between prime(n) and prime(n) + sqrt(prime(n)). Easily checked for prime(n)<1.1e15 in existing maximal gap tables

%D R. K. Guy: Unsolved problems in number theory, 2nd ed., Springer-Verlag,1994; Sections A8, A 9.

%D Paulo Ribenboim: The little book of big primes, Springer-Verlag,1991; 142ff

%H T. D. Noe, <a href="/A058188/b058188.txt">Table of n, a(n) for n = 1..10000</a>

%e a(12) = 2 because between p(12)= 37 and 37+sqrt(37) = 43.08 there are two primes: 41 and 43

%t Table[PrimePi[p+Sqrt[p]]-PrimePi[p],{p,Prime[Range[100]]}] (* _Harvey P. Dale_, Mar 13 2023 *)

%o (PARI) a(n) = my(p=prime(n)); primepi(p+sqrtint(p)) - n; \\ _Michel Marcus_, Jun 21 2017

%Y Cf. A030296.

%K nonn

%O 1,12

%A _Adam Kertesz_, Dec 04 2000