A379444 a(n) is the difference between the least prime > (n+1)^2 and the largest prime < n^2, divided by 2.

4, 5, 8, 7, 11, 10, 11, 11, 15, 18, 17, 15, 17, 17, 21, 24, 25, 21, 23, 24, 31, 27, 30, 29, 30, 30, 40, 34, 40, 39, 35, 38, 38, 37, 41, 40, 42, 45, 48, 54, 51, 51, 47, 56, 50, 51, 57, 52, 66, 57, 60, 57, 64, 57, 65, 71, 65, 69, 67, 64, 78, 66, 68, 69, 72, 77, 81

Offset: 2

Author

Hugo Pfoertner, Dec 23 2024

Keywords

• nonn
• easy

Comments

2*a(n) would be the gap needed between consecutive primes to provide a counterexample to Legendre's conjecture that there is always a prime between n^2 and (n+1)^2. The gaps actually observed are significantly smaller; see A378904 for comparison.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001223, A007491, A053001, A058043, A350100, A378904.

Programs

Mathematica

a[n_]:=(NextPrime[(n+1)^2] - NextPrime[n^2,-1])/2; Array[a,67,2] (* Stefano Spezia, Jan 24 2025 *)

PARI

a379444(n) = (nextprime((n+1)^2) - precprime(n^2))/2

Formula

a(n) = (A007491(n+1) - A053001(n))/2.

a(n) >= n + 2.