A276357 Primes of the form (p*2^x-1)/3, where p is also prime and x is a positive integer.

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 89, 97, 101, 109, 127, 131, 137, 149, 151, 157, 167, 179, 181, 197, 211, 229, 239, 241, 257, 269, 277, 281, 307, 311, 347, 349, 379, 389, 397, 409, 421, 431, 439, 449, 461, 467, 479, 509, 547, 571, 577, 587

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..59.

Wikipedia, Collatz conjecture

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

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).

Example

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.

Mathematica

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A059645 A090190 A325143 * A065041 A065393 A179740

Adjacent sequences: A276354 A276355 A276356 * A276358 A276359 A276360

Keyword

nonn

Author

Michael Cader Nelson, Aug 31 2016

Extensions

Corrected and extended by Michel Marcus, Sep 01 2016

Status

approved