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

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

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice
• easy

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 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.

Sequence gives primes solutions to 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 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

The subset of A055600 which are prime. - Robert G. Wilson v, Jul 19 2014

Named after the American mathematician James Pierpont (1866-1938). - Amiram Eldar, Jun 09 2021

Examples

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

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, section A18, p. 66.
• George E. Martin, Geometric Constructions, Springer, 1998. ISBN 0-387-98276-0.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..8396 (terms 1..795 from T. D. Noe, terms 796..1602 from Joerg Arndt)
• Claudi Alsina and Roger B. Nelson, A Panoply of Polygons, Dolciani Math. Expeditions Vol. 58, AMS/MAA (2023), see page 112.
• Chris K. Caldwell, The Prime Pages.
• David A. Cox and Jerry Shurman, Geometry and number theory on clovers, Amer. Math. Monthly, Vol. 112, No. 8 (2005), pp. 682-704.
• Andrew M. Gleason, Angle Trisection, the Heptagon and the Triskaidecagon, American Mathematical Monthly, Vol. 95, No. 3 (1988), pp. 185-194.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• Joel C. Langer and David A. Singer, Subdividing the trefoil by origami, Geometry, Vol. 2013 (Hindawi Publishing Company, 2013), Article ID 897320. - From _N. J. A. Sloane_, Feb 08 2013
• James Pierpont, On an Undemonstrated Theorem of the Disquisitiones Arithmeticae, American Mathematical Society Bulletin, Vol. 2, No. 3 (1895-1896), pp. 77-83.
• Eric Weisstein's World of Mathematics, Pierpont Prime.
• Index entries for sequences related to the Erdos-Selfridge classification

Crossrefs

Cf. A048135, A048136, A056637, A077497, A077498, A077500, A081426, A122259, A019434, A000668, A000040, A003586, A374577, A374578.

Programs

GAP

K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,3]  then Add(C,Position(B,i)); fi; od;
A005109:=Concatenation([2],List(C,i->A[i])); # Muniru A Asiru, Sep 10 2017

Magma

[p: p in PrimesUpTo(10^8) | forall{d: d in PrimeDivisors(p-1) | d le 3}]; // Bruno Berselli, Sep 24 2012

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; 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 *)

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: */
A005109_upto(lim=1e10)={my(L=List(), k2=1);
until ( lim <= k2 *= 2, my(k23 = k2);
    until ( lim <= k23 *= 3, isprime(k23+1) && listput(L, k23+1));
); Set(L) } /* Joerg Arndt, Sep 22 2012, edited by M. F. Hasler, Mar 17 2024 */

PARI

N=10^8; default(primelimit, N);
print1("2, 3, ");forprime(p=5,N,if(omega(p-1)==3-p%3,print1(p", "))) \\ Chris Boyd, Mar 22 2014

Python

from itertools import islice
from sympy import nextprime
def A005109_gen(): # generator of terms
    p = 2
    while True:
        q = p-1
        q >>= (~q&q-1).bit_length()
        a, b = divmod(q,3)
        while not b:
            a, b = divmod(q:=a,3)
        if q==1:
            yield p
        p = nextprime(p)
A005109_list = list(islice(A005109_gen(),30)) # Chai Wah Wu, Mar 17 2023

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

a(n) = 2^A374577(n) * 3^A374578(n) + 1. - Amiram Eldar, Sep 02 2024

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