A245730 Primes of the form 1+2^k+2^(2*k)+...+2^((j-1)*k) for some k>0, j>0.

3, 5, 7, 17, 31, 73, 127, 257, 8191, 65537, 131071, 262657, 524287, 2147483647, 4432676798593, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Author

Chai Wah Wu, Jul 30 2014

Keywords

• nonn

Comments

Contains the Mersenne primes A000668 which correspond to k=1. In base 2, primes with n 1's and k-1 0's between pairs of 1's. Is a factor of 2^(n*k)-1.

Primes of the form (2^(n*k)-1)/(2^k-1). k=1 gives Mersenne primes 2^n-1 for n in A000043. n=2 and k=2^m gives Fermat primes 2^(2^m)+1 (A019434) for m = 0 to 4. k=n gives (2^(n^2)-1)/(2^n-1) which is prime for n = 2, 3, 7, 59 (A156585, n must be prime). The only other term below 2000 digits is 262657 for k=9 and n=3. - Jens Kruse Andersen, Aug 02 2014

The case n=3 gives the primes in A051154. - John Blythe Dobson

Wells Johnson (1977), 199, Corollary 6, proved that members of this sequence cannot be Wieferich primes (A001220). - John Blythe Dobson

Examples

			The number 4432676798593 is in the list as it is prime and it is equal to 1+2^7+2^(2*7)+2^(3*7)+2^(4*7)+2^(5*7)+2^(6*7).
		

References

• Wells Johnson, On the nonvanishing of Fermat quotients (mod p), J. für Math. 292 (1977), 196-200.

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..25

Crossrefs

Cf. A000043, A000668, A019434, A156585.

Programs

Python

from sympy import isprime
sorted([int(('0'*m+'1')*n,2) for m in range(50) for n in range(1,50) if isprime(int(('0'*m+'1')*n,2))])