

A005109


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


50



2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993, 1769473, 1990657
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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 A005105 for the definition of class r+ primes.
Gleason, p. 191: a regular polygon of n sides can be constructed by ruler, compass and angletrisector 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.
Sequence gives primes solutions to p==+1 (mod phi(p1)).  Benoit Cloitre, Feb 22 2002
These are the primes p for which p1 is 3smooth. Primes for which either p+1 or p1 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 terms p > 3, omega(p1) = 3  p mod 3. Consider terms > 3. Clearly, t > 0. If p == 1 mod 3, u > 0: hence omega(p1) = 2 because p1 has two prime factors. If p == 2 mod 3, u = 0: hence omega(p1) = 1 because p1 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(p1) = 3  p mod 3, p is a term.  Chris Boyd, Mar 22 2014
The subset of A055600 which are prime.  Robert G. Wilson v, Jul 19 2014


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, A18.
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
George E. Martin: Geometric Constructions. Springer, 1998. ISBN 0387982760.
James Pierpont: On an Undemonstrated Theorem of the Disquisitiones Aritmeticae. American Mathematical Society Bulletin 2 (18951896) 77  83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Joerg Arndt, T. D. Noe and Robert G. Wilson v, Table of n, a(n) for n = 1..8396 (the first 795 terms from T. D. Noe, terms to 1602 from Joerg Arndt).
C. K. Caldwell, The Prime Pages
D. A. Cox and J. Shurman, Geometry and number theory on clovers, Amer. Math. Monthly, 112 (2005), 682704.
Andrew M. Gleason, Angle Trisection, the Heptagon and the Triskaidecagon, American Mathematical Monthly, 95 (1988), 185  194.
Eric Weisstein's World of Mathematics, Pierpont Prime


FORMULA

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
{primes p: A126805(PrimePi(p)) = 1}.  R. J. Mathar, Sep 24 2012


EXAMPLE

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


MATHEMATICA

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]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]]  3; Prime[ Select[ Range[3, 6300], ClassMinusNbr[ Prime[ # ]] == 1 &]]
Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ #  1] < 5 &] (* Ray Chandler, Nov 01 2005 *)
mx = 2*10^6; Sort@ Flatten@ Table[2^i*3^j + 1, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}] (* Robert G. Wilson v, Jul 16 2014 *)


PROG

(PARI)
N=10^8; default(primelimit, N);
pq(p)={p=1; (p/(2^valuation(p, 2)*3^valuation(p, 3)))==1; }
forprime(p=2, N, if(pq(p), print1(p, ", ")));
/* Joerg Arndt, Sep 22 2012 */
(PARI) /* much more efficient: */
lim=10^100; x2=0; x3=0; k2=1; k23=1;
{ while ( k2 < lim,
k23 = k2;
while ( k23 < lim,
if ( isprime(k23+1), print(k23+1) );
k23 *= 3;
);
k2 *= 2;
); }
/* Joerg Arndt, Sep 22 2012 */
(MAGMA) [p: p in PrimesUpTo(10^8)  forall{d: d in PrimeDivisors(p1)  d le 3}]; // Bruno Berselli, Sep 24 2012
(PARI)
N=10^8; default(primelimit, N);
print1("2, 3, "); forprime(p=5, N, if(omega(p1)==3p%3, print1(p", "))) \\ Chris Boyd, Mar 22 2014


CROSSREFS

Cf. A048135, A048136, A056637, A005105, A005110, A005111, A005112, A077497, A077498, A077500, A081424, A081425, A081426, A081427, A081428, A081429, A081430, A122259, A019434, A000668, A000040, A003586.
Sequence in context: A109461 A138539 A090422 * A247980 A234851 A179336
Adjacent sequences: A005106 A005107 A005108 * A005110 A005111 A005112


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane, Simon Plouffe


EXTENSIONS

Comments and additional references from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr). More terms from David W. Wilson
More terms from Benoit Cloitre, Feb 22 2002
More terms from Robert G. Wilson v, Mar 20 2003


STATUS

approved



