A075408 Perfect powers pp such that pp+1 is prime.

1, 4, 16, 36, 100, 196, 256, 400, 576, 676, 1296, 1600, 2916, 3136, 4356, 5476, 7056, 8100, 8836, 12100, 13456, 14400, 15376, 15876, 16900, 17956, 21316, 22500, 24336, 25600, 28900, 30976, 32400, 33856, 41616, 42436, 44100, 50176, 52900, 55696

Offset: 1

Author

Zak Seidov, Oct 11 2002

Keywords

• easy
• nonn

Comments

Of 1110 pp's < 10^6, 112 are such that pp+1 is prime and only seven are such that pp-1 is prime (see Mersenne primes (A000668)).

Examples

			pp=324900 is OK because pp=570^2 and pp+1=324901 (prime).
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597 (perfect powers), A072868 (pp-1 is prime).

Cf. A002496.

Programs

Mathematica

pp = Join[ Select[ Range[56000], Apply[GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]]; Select[pp, PrimeQ[ # + 1] & ]

Python

from sympy import isprime
print([n**2 for n in range(1,240) if isprime(n**2 + 1)]) # Karl-Heinz Hofmann, Feb 02 2023

Formula

a(n) = A002496(n) - 1.

Extensions

Edited by Robert G. Wilson v, Oct 14 2002

Edited by N. J. A. Sloane, Dec 17 2009 at the suggestion of Rick Shepherd