

A007491


Smallest prime > n^2.
(Formerly M1389)


31



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
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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.  JeanChristophe 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>inf} 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

T. D. Noe and JeanChristophe 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


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 JeanChristophe Hervé, Oct 26 2013


STATUS

approved



