A005105 Class 1+ primes: primes of the form 2^i*3^j - 1 with i, j >= 0. (Formerly M0665)

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

1,1

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

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

Formula

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

Example

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.

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 &]]

Prog

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

Crossrefs

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

Sequence in context: A038953 A237288 A293074 * A086566 A235213 A188552

Adjacent sequences: A005102 A005103 A005104 * A005106 A005107 A005108

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Benoit Cloitre, Feb 22 2002

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

Status

approved