

A276357


Primes of the form (p*2^x1)/3, where p is also prime and x is a positive integer.


0



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
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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*2^11)/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



