A079636 Smallest number whose reciprocal fits in the square-root gap of consecutive primes.

4, 2, 3, 2, 4, 2, 5, 3, 2, 6, 2, 4, 7, 4, 3, 3, 8, 3, 5, 9, 3, 5, 4, 3, 5, 11, 6, 11, 6, 2, 6, 4, 12, 3, 13, 5, 5, 7, 5, 5, 14, 3, 14, 7, 15, 3, 3, 8, 16, 8, 6, 16, 4, 6, 6, 6, 17, 6, 9, 17, 4, 3, 9, 18, 9, 3, 7, 4, 19, 10, 7, 5, 7, 7, 10, 7, 5, 10, 6, 5, 21

Offset: 1

Author

Rainer Rosenthal, Jan 30 2003

Keywords

• easy
• nonn

Comments

Is the limit of sqrt(P_(n+1)) - sqrt(P_n) = 0?

Examples

			a(3) = 3 because p(3)=5, p(4)=7, w=sqrt(5), w'=sqrt(7) and 1/(w'-w)=2.44.
		

References

• Jim Ferry, sci.math, Jan 30 2003

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A252477.

Programs

Maple

a:= n-> ((w, v)-> ceil(1/(w-v)))(map(sqrt@ithprime, [n+1, n])[]):
seq(a(n), n=1..81);  # Alois P. Heinz, Aug 23 2025

Mathematica

Ceiling[1/Subtract @@@ Reverse[Partition[Sqrt[Prime[Range[100]]], 2, 1], 2]] (* Paolo Xausa, Aug 24 2025 *)

Formula

a(n) = ceiling(1/(w'-w)) where w=sqrt(p(n)) and w'=sqrt(p(n+1)).

a(n) = A252477(n) + 1. - Hugo Pfoertner, Aug 23 2025

Extensions

More terms from Sean A. Irvine, Aug 23 2025