%I #38 Jul 03 2020 23:36:33

%S 3,5,7,11,13,17,19,29,31,37,41,47,53,59,61,67,71,89,97,101,109,127,

%T 131,137,149,151,157,167,179,181,197,211,229,239,241,257,269,277,281,

%U 307,311,347,349,379,389,397,409,421,431,439,449,461,467,479,509,547,571,577,587

%N Primes of the form (p*2^x-1)/3, where p is also prime and x is a positive integer.

%C Relationship to Collatz (3x+1) problem: when one of these primes appears in a hailstone sequence, the next odd number in the sequence must be prime. - _Michael Cader Nelson_, Jul 03 2020

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%F The value of p is (3*a(n)+1)/2^x as well as the respective term in A087273 evaluated for a(n), while the value of x is the related exponent in A087963 unless 3*a(n)+1 is a power of 2 (e.g., n = 1).

%e 3 is in the sequence because 3 = (5*2^1-1)/3 and both 3 and 5 are prime numbers; while 23 is not in the sequence because the only positive integer values (p,x) to give 23 are (35,1) and 35 is not prime.

%t mx = 590; Select[ Sort@ Flatten@ Table[(Prime[p]*2^x - 1)/3, {x, Log2[mx/3]}, {p, PrimePi[3 mx/2^x]}], PrimeQ] (* _Robert G. Wilson v_, Nov 01 2016 *)

%o (PARI) lista(nn) = {forprime(p=2, nn, z = 3*p+1; x = valuation(z, 2); for (ex = 1, x, if (isprime(z/2^ex), print1(p, ", "); break;);););} \\ _Michel Marcus_, Sep 01 2016

%Y Cf. A087273, A087963. A177330 (lists all exponents x).

%K nonn

%O 1,1

%A _Michael Cader Nelson_, Aug 31 2016

%E Corrected and extended by _Michel Marcus_, Sep 01 2016