A144831 (n+1)^2 - (smallest prime > n^2).

2, 4, 5, 8, 7, 12, 11, 14, 17, 20, 17, 20, 23, 28, 29, 32, 31, 30, 33, 40, 41, 42, 35, 48, 45, 52, 51, 54, 47, 54, 57, 58, 65, 62, 67, 72, 71, 74, 77, 80, 71, 72, 75, 76, 89, 80, 91, 92, 89, 98, 95, 102, 97, 108, 99, 112, 113, 110, 109, 114, 117, 122, 107, 126, 127, 132, 131

Offset: 1

Author

Enoch Haga, Sep 21 2008

Keywords

• nonn
• easy

Comments

Suggested by Conjecture 60 in Carlos Rivera's The Prime Puzzles & Problems Connection.

Legendre's conjecture that there is always a prime between n^2 and (n+1)^2 is equivalent to a(n) >= 0 for all n. As the conjecture is still opened, it is not proved that a(n) is nonn, although the keyword is automatically added. - Jean-Christophe Hervé, Oct 26 2013

Examples

			a(2)=4 because n=2, 2^2=4 and (2+1)^2=9. The gap in which primes are to be found is 4 - 9. Next prime=5 and 9-5=4. For a(3)=5, 3^2=9 and (3+1)^2=16. Next prime=11 and 16-11=5.
		

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007491, A053000, A144832, A144256, A014085.

Programs

Mathematica

Table[n^2-NextPrime[(n-1)^2],{n,2,70}] (* Harvey P. Dale, Jan 22 2019 *)

PARI

a(n) = (n+1)^2 - nextprime(n^2); \\ Michel Marcus, Jun 08 2014

Formula

Calculate n^2 and (n+1)^2, e.g. 4 - 9. Find the next prime following n^2 and subtract from (n+1)^2. Next prime is 5 so 9-5=4, the distance from next prime to (n+1)^2.

a(n) = (n+1)^2 - A007491(n).

Extensions

Definition rewritten by N. J. A. Sloane, Sep 28 2008

Definition rewritten by Jean-Christophe Hervé, Oct 26 2013