A245730 - OEIS

%I #16 Aug 24 2014 17:59:11

%S 3,5,7,17,31,73,127,257,8191,65537,131071,262657,524287,2147483647,

%T 4432676798593,2305843009213693951,618970019642690137449562111,

%U 162259276829213363391578010288127,170141183460469231731687303715884105727

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

%C 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.

%C 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

%C The case n=3 gives the primes in A051154. - _John Blythe Dobson_

%C Wells Johnson (1977), 199, Corollary 6, proved that members of this sequence cannot be Wieferich primes (A001220). - _John Blythe Dobson_

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

%H Jens Kruse Andersen, <a href="/A245730/b245730.txt">Table of n, a(n) for n = 1..25</a>

%e 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).

%o (Python) from sympy2 import isprime

%o 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))])

%Y Cf. A000043, A000668, A019434, A156585.

%K nonn

%O 1,1

%A _Chai Wah Wu_, Jul 30 2014