cp's OEIS Frontend

Conjecture: a(n) is ~ (n*log(2))^2/9 as n increases.

For n from 1 to 2000, a(n)/prime(n) is always < 1.8.

As N increases, (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) tends to log(2)/3; this is consistent with the prime number theorem as the probability that x*2^n-1 is prime with odd x divisible by 3 is ~ 3/(n*log(2)) and after n*log(2)/3 try (n*log(2)/3)*(3/(n*log(2))) = 1.

For some n (6*k*n-3)*2^n-1 is composite for any k.

For n=15+20*j, n=7+21*j, n=77+110*j, n=26+156*j, n=266+342*j, n=261+812*j, n=2368+1332*j, n=477+2756*j, n=2183+3422*j and more others (6*k*n-3)*2^n-1 is always composite for any k and any j.

For n=4390+187892*j, (6*k*n-3)*2^n-1 is always divisible by one of the 82 primes between 5 and 443, 4390=10*439 and 187892=438*439.

For n=6152+596744*j, (6*k*n-3)*2^n-1 is always divisible by one of the 134 primes between 3 and 773, 6152=8*769 and 596744=768*769.

For n=11*1229+1228*1229*j, (6*n*k-3)*2^n-1 is always divisible by one of the 199 primes between 3 and 1231 except 11.

For n=27*1399+1398*1399*j, (6*n*k-3)*2^n-1 is always divisible by one of the 220 primes between 3 and 1409.

For n=5*11*1619+1618*1619*j, (6*n*k-3)*2^n-1 is always divisible by one of the 253 primes between 5 and 1621 except 11.