A268175 Smallest number k such that k*(2^(2*A000043(n))-1)+1 is prime.

2, 2, 4, 6, 12, 82, 14, 22, 244, 44, 120, 94, 1010, 764, 834, 1076, 516, 3252, 1384, 1664, 7040, 6104, 20942, 14344, 37142, 12522, 12554, 64160, 32172, 44460, 49400, 291726

Offset: 1

Author

Pierre CAMI, Jan 28 2016

Keywords

• nonn
• more
• hard

Comments

The numbers k*(2^(2*A000043(n))-1)+1 may be written as k*(2^A000043(n)-1)*(2^A000043(n)+1)+1 or k*Mersenne(n)*(Mersenne(n)+2)+1 and so may be proved primes.

All the numbers a(n)*(2^(2*A000043(n))-1)+1 for n=1 to 32 have been proved to be primes.

Examples

			2*(2^(2*2)-1)+1 = 31 (prime) and A000043(1) = 2, so a(1) = 2.
2*(2^(2*3)-1)+1 = 127 (prime) and A000043(2) = 3, so a(2) = 2.
		

Crossrefs

Cf. A000043.

Programs

Mathematica

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657};
lst = {}; maxk = 5000; maxn = 15;
For[n = 1, n ≤ maxn, n++,
  For[k = 0, k ≤ maxk, k++,
    If[PrimeQ[k*(2^(2*A000043[[n]]) - 1) + 1], AppendTo[lst, k]; Break[]]
  ]
];
lst (* Robert Price, Apr 05 2016 *)