A007491 Smallest prime > n^2.

2, 5, 11, 17, 29, 37, 53, 67, 83, 101, 127, 149, 173, 197, 227, 257, 293, 331, 367, 401, 443, 487, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1031, 1091, 1163, 1229, 1297, 1373, 1447, 1523, 1601, 1693, 1777, 1861, 1949, 2027, 2129, 2213, 2309, 2411, 2503

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v, R. K. Guy

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.

Legendre's conjecture is equivalent to a(n) < (n+1)^2. - Jean-Christophe Hervé, Oct 26 2013

From Jaroslav Krizek, Apr 02 2016: (Start)

Conjectures:

1) There is always a prime p between n^2 and n^2+n (verified up to 13*10^6).

2) a(n) is the smallest prime p such that n^2 < p < n^2+n; a(n) < n^2+n.

3) For all numbers k >= 1 there is the smallest number m > 2*(k+1) such that for all numbers n >= m there is always a prime p between n^2 and n^2 + n - 2k. Sequence of numbers m for k >= 1: 6, 8, 12, 13, 14, 24, 24, 24, 30, 30, 30, 31, 33, 35, 43, ...; lim_{k->oo} m/2k = 1. Example: k=2; for all numbers n >= 8 there is always a prime p between n^2 and n^2 + n - 4. (End)

References

• Archimedeans Problems Drive, Eureka, 24 (1961), 20.
• J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 19.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Landau's Problem.
• Eric Weisstein's World of Mathematics, Legendre's Conjecture.

Crossrefs

Cf. A053000, A053001, A014085, A144831.

Cf. A007918, A000290.

Programs

Haskell

a007491 = a007918 . a000290  -- Reinhard Zumkeller, Jun 07 2015

Magma

[NextPrime(n^2): n in [1..50]]; // Vincenzo Librandi, Apr 30 2015

Maple

[seq(nextprime(i^2), i=1..100)];

Mathematica

NextPrime[Range[60]^2]  (* Harvey P. Dale, Mar 24 2011 *)

PARI

vector(100,i,nextprime(i^2))

Python

from sympy import nextprime
def a(n): return nextprime(n**2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 13 2023

Formula

a(n) = A007918(A000290(n)). - Reinhard Zumkeller, Jun 07 2015

Extensions

More terms from Labos Elemer, Nov 17 2000

Definition modified by Jean-Christophe Hervé, Oct 26 2013