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

2,1

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

Formula

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

a(n) >= n + 2.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A021222 A132023 A378909 * A380457 A107758 A082468

Adjacent sequences: A379441 A379442 A379443 * A379445 A379446 A379447

Keyword

nonn,easy

Author

Hugo Pfoertner, Dec 23 2024

Status

approved