A005105 Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0.

2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73727, 131071, 139967, 165887, 294911, 314927, 442367, 472391, 497663, 524287, 786431, 995327

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

The definition is given by Guy: a prime p is in class 1+ if the only prime divisors of p + 1 are 2 or 3; and p is in class r+ if every prime factor of p + 1 is in some class <= r+ - 1, with equality for at least one prime factor. - N. J. A. Sloane, Sep 22 2012

See A005109 for the definition of class r- primes.

Odd terms are primes satisfying p==-1 (mod phi(p+1)). - Benoit Cloitre, Feb 22 2002

These are the primes p for which p+1 is 3-smooth. Primes for which either p+1 or p-1 have many small factors are more easily proved prime, so most of the largest primes found have this property. - Michael B. Porter, Feb 19 2013

For n>1, x=2*a(n) is a solution to the equation phi(sigma(x)) = x-phi(x). Also all Mersenne primes are in the sequence. - Jahangeer Kholdi, Sep 28 2014

Examples

			23 is in the sequence since 23 is prime and 23 + 1 = 24 = 2^3 * 3 has all prime factors less than or equal to 3.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, A18.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ray Chandler, Table of n, a(n) for n = 1..7170 (terms < 10^1000; terms 1..691 from T. D. Noe, terms 692..5000 from Charles R Greathouse IV)
• C. K. Caldwell, The Prime Pages.
• G. Everest, P. Rogers and T. Ward, A higher-rank Mersenne problem, pp. 95-107 of ANTS 2002, Lect. Notes Computer Sci. 2369 (2002).
• R. J. Mathar, Maple programs to generate b-files for b005105 to b005108, b081633 etc.
• Index entries for sequences related to the Erdos-Selfridge classification

Crossrefs

Cf. A069353, A069356, A005109, A005108, A019434, A000668, A000040, A003586, A129469.

Programs

GAP

A:=Filtered([1..10^7],IsPrime);;     I:=[3];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A005105:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i])); # Muniru A Asiru, Sep 28 2017

Magma

[p: p in PrimesUpTo(6*10^6) | forall{d: d in PrimeDivisors(p+1) | d le 3}]; // Bruno Berselli, Sep 24 2012

Maple

For Maple program see Mathar link.
# Alternative:
N:= 10^6: # to get all terms <= N
select(isprime,{seq(seq(2^i*3^j-1, i=0..ilog2(N/3^j)), j=0..floor(log[3](N)))});
# if using Maple 11 or earlier, uncomment the following line
# sort(convert(%,list));  # Robert Israel, Sep 28 2014

Mathematica

mx = 10^6; Select[ Sort@ Flatten@ Table[2^i*3^j - 1, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}], PrimeQ] (* or *)
Prime[ Select[ Range[78200], Mod[ Prime[ # ] + 1, EulerPhi[ Prime[ # ] + 1]] == 0 &]] (* or *)
PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] + 1]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[3, 78200], ClassPlusNbr[ Prime[ # ]] == 1 &]]

PARI

list(lim)=my(v=List(), N); lim=1+lim\1; for(n=0, logint(lim,3), N=3^n; while(N<=lim, if(ispseudoprime(N-1),listput(v, N-1)); N<<=1)); Set(v) \\ Charles R Greathouse IV, Jul 15 2011; corrected Sep 22 2015

Python

from itertools import count, islice
from sympy import integer_log, isprime
def A069353(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x+1)//3**i).bit_length() for i in range(integer_log(x+1,3)[0]+1))
    return bisection(f,n-1,n-1)
def A005105_gen(): # generator of terms
    return filter(lambda n:isprime(n), map(A069353,count(1)))
A005105_list = list(islice(A005105_gen(),30)) # Chai Wah Wu, Mar 31 2025

Formula

{primes p : A126433(PrimePi(p)) = 1 }. - R. J. Mathar, Sep 24 2012

Extensions

More terms from Benoit Cloitre, Feb 22 2002

Edited and extended by Robert G. Wilson v, Mar 20 2003