A053000 a(n) = (smallest prime > n^2) - n^2.

2, 1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 5, 4, 1, 2, 1, 4, 7, 6, 1, 2, 3, 12, 1, 6, 1, 4, 3, 12, 7, 6, 7, 2, 7, 4, 1, 4, 3, 2, 1, 12, 13, 12, 13, 2, 13, 4, 5, 10, 3, 8, 3, 10, 1, 12, 1, 2, 7, 10, 7, 6, 3, 20, 3, 4, 1, 4, 13, 22, 3, 10, 5, 4, 1, 14, 3, 10, 5, 6, 21, 2, 9, 10, 1, 4, 15, 4, 9, 6, 1, 6, 3, 14

Offset: 0

Author

N. J. A. Sloane, Feb 21 2000

Keywords

• nonn
• easy
• nice

Comments

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.

Record values are listed in A070317, their indices in A070316. - M. F. Hasler, Mar 23 2013

Conjecture: a(n) <= 1+phi(n) = 1+A000010(n), for n>0. This improves on Oppermann's conjecture, which says a(n) < n. - Jianglin Luo, Sep 22 2023

References

• J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
• R. K. Guy, Unsolved Problems in Number Theory, Section A1.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A007491, A013632, A053001, A014085, A070316, A085099, A058055, A069003.

Programs

Magma

[NextPrime(n^2) - n^2: n in [0..100]]; // Vincenzo Librandi, Jul 06 2015

Maple

A053000 := n->nextprime(n^2)-n^2;

Mathematica

nxt[n_]:=Module[{n2=n^2},NextPrime[n2]-n2]
nxt/@Range[0,100]  (* Harvey P. Dale, Dec 20 2010 *)

PARI

A053000(n)=nextprime(n^2)-n^2  \\ M. F. Hasler, Mar 23 2013

Python

from sympy import nextprime
def a(n): nn = n*n; return nextprime(nn) - nn
print([a(n) for n in range(94)]) # Michael S. Branicky, Feb 17 2022

Formula

a(n) = A013632(n^2). - Robert Israel, Jul 06 2015

Extensions

More terms from James Sellers, Feb 22 2000