A007491 Smallest prime > n^2. (Formerly M1389)

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

1,1

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.

Formula

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

Maple

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

Mathematica

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

Prog

(PARI) vector(100, i, nextprime(i^2))
(Magma) [NextPrime(n^2): n in [1..50]]; // Vincenzo Librandi, Apr 30 2015
(Haskell)
a007491 = a007918 . a000290  -- Reinhard Zumkeller, Jun 07 2015
(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

Crossrefs

Cf. A053000, A053001, A014085, A144831.

Cf. A007918, A000290.

Sequence in context: A023222 A289250 A278049 * A124850 A156850 A156611

Adjacent sequences: A007488 A007489 A007490 * A007492 A007493 A007494

Keyword

nonn,easy,nice

Author

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

Extensions

More terms from Labos Elemer, Nov 17 2000

Definition modified by Jean-Christophe Hervé, Oct 26 2013

Status

approved