A005105 - OEIS

%I M0665 #81 Mar 27 2024 08:08:18

%S 2,3,5,7,11,17,23,31,47,53,71,107,127,191,383,431,647,863,971,1151,

%T 2591,4373,6143,6911,8191,8747,13121,15551,23327,27647,62207,73727,

%U 131071,139967,165887,294911,314927,442367,472391,497663,524287,786431,995327

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

%C 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

%C See A005109 for the definition of class r- primes.

%C Odd terms are primes satisfying p==-1 (mod phi(p+1)). - _Benoit Cloitre_, Feb 22 2002

%C 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

%C 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

%D R. K. Guy, Unsolved Problems in Number Theory, A18.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ray Chandler, <a href="/A005105/b005105.txt">Table of n, a(n) for n = 1..7170</a> (terms < 10^1000; terms 1..691 from T. D. Noe, terms 692..5000 from Charles R Greathouse IV)

%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/">The Prime Pages</a>.

%H G. Everest, P. Rogers and T. Ward, <a href="https://ueaeprints.uea.ac.uk/19707/">A higher-rank Mersenne problem</a>, pp. 95-107 of ANTS 2002, Lect. Notes Computer Sci. 2369 (2002).

%H R. J. Mathar, <a href="/A005105/a005105.txt">Maple programs to generate b-files for b005105 to b005108, b081633 etc.</a>

%H <a href="/index/Pri#primes_Erdos_Selfridge">Index entries for sequences related to the Erdos-Selfridge classification</a>

%F {primes p : A126433(PrimePi(p)) = 1 }. - _R. J. Mathar_, Sep 24 2012

%e 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.

%p For Maple program see Mathar link.

%p # Alternative:

%p N:= 10^6: # to get all terms <= N

%p select(isprime,{seq(seq(2^i*3^j-1, i=0..ilog2(N/3^j)), j=0..floor(log[3](N)))});

%p # if using Maple 11 or earlier, uncomment the following line

%p # sort(convert(%,list));  # _Robert Israel_, Sep 28 2014

%t 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 *)

%t Prime[ Select[ Range[78200], Mod[ Prime[ # ] + 1, EulerPhi[ Prime[ # ] + 1]] == 0 &]] (* or *)

%t 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 &]]

%o (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

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

%o (GAP)

%o A:=Filtered([1..10^7],IsPrime);;     I:=[3];;

%o B:=List(A,i->Elements(Factors(i+1)));;

%o C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;

%o 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

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Benoit Cloitre_, Feb 22 2002

%E Edited and extended by _Robert G. Wilson v_, Mar 20 2003