A058188 Number of primes between prime(n) and prime(n) + sqrt(prime(n)), where prime(n) is the n-th prime.

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, 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, 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

Offset: 1

Author

Adam Kertesz, Dec 04 2000

Keywords

• nonn

Comments

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

Examples

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

References

• R. K. Guy: Unsolved problems in number theory, 2nd ed., Springer-Verlag,1994; Sections A8, A 9.
• Paulo Ribenboim: The little book of big primes, Springer-Verlag,1991; 142ff

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A030296.

Programs

Mathematica

Table[PrimePi[p+Sqrt[p]]-PrimePi[p],{p,Prime[Range[100]]}] (* Harvey P. Dale, Mar 13 2023 *)

PARI

a(n) = my(p=prime(n)); primepi(p+sqrtint(p)) - n; \\ Michel Marcus, Jun 21 2017