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 A005109 Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1. (Formerly M0673) 54

%I M0673

%S 2,3,5,7,13,17,19,37,73,97,109,163,193,257,433,487,577,769,1153,1297,

%T 1459,2593,2917,3457,3889,10369,12289,17497,18433,39367,52489,65537,

%U 139969,147457,209953,331777,472393,629857,746497,786433,839809,995329,1179649,1492993,1769473,1990657

%N Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.

%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 A005105 for the definition of class r+ primes.

%C Gleason, p. 191: a regular polygon of n sides can be constructed by ruler, compass and angle-trisector iff n = 2^r * 3^s * p_1 * p_2 .... p_k, where p_1, p_2,....,p_k are distinct elements of this sequence and >3.

%C Sequence gives primes solutions to 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 terms p > 3, omega(p-1) = 3 - p mod 3. Consider terms > 3. Clearly, t > 0. If p == 1 mod 3, u > 0: hence omega(p-1) = 2 because p-1 has two prime factors. If p == 2 mod 3, u = 0: hence omega(p-1) = 1 because p-1 is a power of 2. The latter case corresponds to terms that are Fermat primes > 3. Similar arguments demonstrate the converse, that for p > 3, if omega(p-1) = 3 - p mod 3, p is a term. - _Chris Boyd_, Mar 22 2014

%C The subset of A055600 which are prime. - _Robert G. Wilson v_, Jul 19 2014

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

%D J. C. Langer and D. A. Singer, Subdividing the Trefoil by Origami, Geometry (Hindawi Publishing Company), 2013, #ID 897320. - From _N. J. A. Sloane_, Feb 08 2013

%D George E. Martin: Geometric Constructions. Springer, 1998. ISBN 0-387-98276-0.

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

%H Robert G. Wilson v, <a href="/A005109/b005109.txt">Table of n, a(n) for n = 1..8396</a> (terms 1..795 from T. D. Noe, terms 796..1602 from Joerg Arndt)

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

%H D. A. Cox and J. Shurman, <a href="http://www.jstor.org/stable/30037571">Geometry and number theory on clovers</a>, Amer. Math. Monthly, 112 (2005), 682-704.

%H Andrew M. Gleason, <a href="http://www.jstor.org/stable/2323624">Angle Trisection, the Heptagon and the Triskaidecagon</a>, American Mathematical Monthly, 95 (1988), 185 - 194.

%H James Pierpont, <a href="https://doi.org/10.1090/S0002-9904-1895-00317-1">On an Undemonstrated Theorem of the Disquisitiones Arithmeticae</a>, American Mathematical Society Bulletin 2 (1895-1896) pp. 77-83.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierpontPrime.html">Pierpont Prime</a>

%F A122257(a(n)) = 1; A122258(n) = number of Pierpont primes <= n; A122260 gives numbers having only Pierpont primes as factors. - _Reinhard Zumkeller_, Aug 29 2006

%F {primes p: A126805(PrimePi(p)) = 1}. - _R. J. Mathar_, Sep 24 2012

%e 97 = 2^5*3 + 1 is a member.

%t PrimeFactors[n_Integer] := Flatten[ Table[ #[], {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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[3, 6300], ClassMinusNbr[ Prime[ # ]] == 1 &]]

%t Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* _Ray Chandler_, Nov 01 2005 *)

%t mx = 2*10^6; Select[Sort@ Flatten@ Table[2^i*3^j + 1, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}], PrimeQ] (* _Robert G. Wilson v_, Jul 16 2014, edited by _Michael De Vlieger_, Aug 23 2017 *)

%o (PARI)

%o N=10^8; default(primelimit,N);

%o pq(p)={p-=1; (p/(2^valuation(p,2)*3^valuation(p,3)))==1;}

%o forprime(p=2,N,if(pq(p),print1(p,", ")));

%o /* _Joerg Arndt_, Sep 22 2012 */

%o (PARI) /* much more efficient: */

%o lim=10^100; x2=0; x3=0; k2=1; k23=1;

%o { while ( k2 < lim,

%o k23 = k2;

%o while ( k23 < lim,

%o if ( isprime(k23+1), print(k23+1) );

%o k23 *= 3;

%o );

%o k2 *= 2;

%o ); }

%o /* _Joerg Arndt_, Sep 22 2012 */

%o (MAGMA) [p: p in PrimesUpTo(10^8) | forall{d: d in PrimeDivisors(p-1) | d le 3}]; // _Bruno Berselli_, Sep 24 2012

%o (PARI)

%o N=10^8; default(primelimit, N);

%o print1("2, 3, ");forprime(p=5,N,if(omega(p-1)==3-p%3,print1(p", "))) \\ _Chris Boyd_, Mar 22 2014

%o (GAP)

%o K:=10^7;; # to get all terms <= K.

%o A:=Filtered([1..K],IsPrime);;

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

%o C:=[];; for i in B do if Elements(i)= or Elements(i)=[2,3] then Add(C,Position(B,i)); fi; od;

%o A005109:=Concatenation(,List(C,i->A[i])); # _Muniru A Asiru_, Sep 10 2017

%Y Cf. A048135, A048136, A056637, A005105, A005110, A005111, A005112, A077497, A077498, A077500, A081424, A081425, A081426, A081427, A081428, A081429, A081430, A122259, A019434, A000668, A000040, A003586.

%K nonn,nice,easy

%O 1,1

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