This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A245730 #23 Jun 24 2025 10:45:22
%S A245730 3,5,7,17,31,73,127,257,8191,65537,131071,262657,524287,2147483647,
%T A245730 4432676798593,2305843009213693951,618970019642690137449562111,
%U A245730 162259276829213363391578010288127,170141183460469231731687303715884105727
%N A245730 Primes of the form 1+2^k+2^(2*k)+...+2^((j-1)*k) for some k>0, j>0.
%C A245730 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 A245730 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 A245730 The case n=3 gives the primes in A051154. - _John Blythe Dobson_
%C A245730 Wells Johnson (1977), 199, Corollary 6, proved that members of this sequence cannot be Wieferich primes (A001220). - _John Blythe Dobson_
%D A245730 Wells Johnson, On the nonvanishing of Fermat quotients (mod p), J. für Math. 292 (1977), 196-200.
%H A245730 Jens Kruse Andersen, <a href="/A245730/b245730.txt">Table of n, a(n) for n = 1..25</a>
%e A245730 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 A245730 (Python) from sympy import isprime
%o A245730 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 A245730 Cf. A000043, A000668, A019434, A156585.
%K A245730 nonn
%O A245730 1,1
%A A245730 _Chai Wah Wu_, Jul 30 2014