A053001 Largest prime < n^2.

3, 7, 13, 23, 31, 47, 61, 79, 97, 113, 139, 167, 193, 223, 251, 283, 317, 359, 397, 439, 479, 523, 571, 619, 673, 727, 773, 839, 887, 953, 1021, 1087, 1153, 1223, 1291, 1367, 1439, 1511, 1597, 1669, 1759, 1847, 1933, 2017, 2113, 2207, 2297, 2399, 2477, 2593

Offset: 2

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.

Legendre's conjecture is equivalent to a(n) > (n-1)^2. - John W. Nicholson, Dec 11 2013

References

• J. R. Goldman, The Queen of Mathematics, 1998, p. 82.

Links

• T. D. Noe, Table of n, a(n) for n=2..1000

Crossrefs

Cf. A007491, A053000, A014085.

Cf. A007917, A000290.

Programs

Haskell

a053001 = a007917 . a000290  -- Reinhard Zumkeller, Jun 07 2015

Maple

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

Mathematica

Table[Prime[PrimePi[n^2]], {n, 2, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
Table[NextPrime[n^2, -1], {n, 2, 60}] (* Jean-François Alcover, Oct 14 2013 *)

PARI

a(n) = precprime(n^2) \\ Michel Marcus, Oct 14 2013

Python

from sympy import prevprime
def a(n):  return prevprime(n*n)
print([a(n) for n in range(2, 52)]) # Michael S. Branicky, Jul 29 2022

Formula

a(n) = A007917(A000290(n)). - Reinhard Zumkeller, Jun 07 2015

Extensions

More terms from James Sellers, Feb 22 2000