A087963 -id:A087963 - cp's OEIS frontend

A087964 a(n) is the least prime p such that exponent of highest power of 2 dividing 3p+1 equals n.

3, 17, 13, 5, 53, 149, 1237, 1109, 853, 2389, 3413, 17749, 128341, 70997, 251221, 415061, 218453, 2708821, 27088213, 29709653, 3495253, 85284181, 13981013, 39146837, 794121557, 1498764629, 492131669, 626349397, 13779686741

Offset: 1

Labos Elemer, Sep 18 2003

nonn

			p = 218453 is the first prime so that 3*p+1 = 655360 = (2^18)*5 has 18 as exponent of 2 in 3p+1, thus a(18) = 218453.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A087272, A087273, A087274, A007814, A087230, A087963.

f:= proc(n)
   local m,t,p;
   t:= 2^n;
   for m from 1 + 4*(n mod 2) by 6 do
     p:= (t*m-1)/3;
     if isprime(p) then return p fi
   od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 18 2017

a[n_] := Module[{m, t = 2^n, p}, For[m = 1 + 4 Mod[n, 2], True, m += 6, p = (t m - 1)/3; If[PrimeQ[p], Return[p]]]];
Array[a, 100] (* Jean-François Alcover, Aug 28 2020, after Robert Israel *)

a(n) = A000040(Min{x; A007814(1 + 3*A000040(x)) = n}).

More terms from Ray Chandler, Sep 21 2003

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

Original entry on oeis.org

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

Michael Cader Nelson, Aug 31 2016

nonn

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

			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.

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

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 *)

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

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).

Corrected and extended by Michel Marcus, Sep 01 2016

cp's OEIS Frontend

A087964 a(n) is the least prime p such that exponent of highest power of 2 dividing 3p+1 equals n.

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