A058220 Ultra-useful primes: smallest k such that 2^(2^n) - k is prime.

1, 3, 5, 15, 5, 59, 159, 189, 569, 105, 1557, 2549, 2439, 13797, 25353, 5627, 24317, 231425, 164073

Offset: 1

Author

Warren D. Smith, Nov 30 2000

Keywords

• nonn
• hard
• nice
• more

Comments

2^(2^16) - 5627 was found by Joel Levy in Feb 2004. - Donovan Johnson, Sep 13 2008

Corresponding numbers to entries a(1) to a(12) are proven primes, higher terms are probable primes. - Matthias Baur, Mar 17 2020

Examples

			For n = 3, we see that 2^(2^3) = 2^8 = 256, which is clearly not prime.
256 - 1 = 255 = 3 * 5 * 17, so a(3) is not 1.
256 - 2 = 254 = 2 * 127, so a(3) is not 2 either.
256 - 3 = 253 = 11 * 23, so a(3) is not 3 either.
256 - 5 = 251, which is prime, so a(3) = 5.
		

Links

• Rosetta Code, Ultra-useful primes.

Crossrefs

Cf. A001146, A013603, A058221.

Programs

Mathematica

ultraUseful[n_] := Module[{x = 2^(2^n)}, x - NextPrime[x, -1]]; Array[ultraUseful, 17] (* Harvey P. Dale, Jun 04 2011 *)

Formula

a(n) = A013603(2^n). - Jinyuan Wang, Jun 06 2020

Extensions

a(16)-a(17) from Donovan Johnson, Sep 13 2008

a(18) from Matthias Baur, Mar 17 2020

a(19) from Matthias Baur, Apr 04 2020