

A245730


Primes of the form 1+2^k+2^(2*k)+...+2^((n1)*k) for some k>0, n>0.


3



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

1,1


COMMENTS

Contains the Mersenne primes A000668 which correspond to k=1. In base 2, primes with n 1's and k1 0's between pairs of 1's. Is a factor of 2^(n*k)1.
Primes of the form (2^(n*k)1)/(2^k1). k=1 gives Mersenne primes 2^n1 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^n1) 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
Wells Johnson (1977), 199, Corollary 6, proved that members of this sequence cannot be Wieferich primes (A001220).  John Blythe Dobson


REFERENCES

Wells Johnson, On the nonvanishing of Fermat quotients (mod p), J. für Math. 292 (1977), 196200.


LINKS



EXAMPLE

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).


PROG

(Python) from sympy2 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))])


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



