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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, 2654209, 5038849, 5308417, 8503057, 11337409, 14155777, 19131877, 28311553, 57395629, 63700993, 71663617, 86093443 (list; graph; refs; listen; history; text; internal format)
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 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

REFERENCES

D. A. Cox and J. Shurman, Geometry and number theory on clovers, Amer. Math. Monthly, 112 (2005), 682-704.

Andrew M. Gleason: Angle Trisection, the Heptagon and the Triskaidecagon. American Mathematical Monthly 95 (1988) 185 - 194.

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 0-387-98276-0.

James Pierpont: On an Undemonstrated Theorem of the Disquisitiones Aritmeticae. American Mathematical Society Bulletin 2 (1895-1896) 77 - 83.

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

LINKS

T. D. Noe and Joerg Arndt, Table of n, a(n) for n = 1..1602, these are all terms <= 10^200; the first 795 terms (<=10^100) were computed by T. D. Noe

C. K. Caldwell, The Prime Pages

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

Take[ Select[ Sort[ Flatten[ Table[2^t*3^u + 1, {t, 0, 22}, {u, 0, 16}]]], PrimeQ[ # ] &], 42] (* 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]]; 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 *)

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(p-1) | d le 3}]; // Bruno Berselli, Sep 24 2012

(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

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 * A234851 A179336 A080608

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

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Last modified April 16 18:58 EDT 2014. Contains 240627 sequences.