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A005109
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Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.
(Formerly M0673)
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50
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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
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OFFSET
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1,1
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COMMENTS
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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 proven prime, so most of the largest primes found have this property. - Michael B. Porter, Feb 19 2013
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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97 = 2^5*3 + 1 is a member.
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MATHEMATICA
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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)
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PROG
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(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
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CROSSREFS
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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 * A179336 A080608 A137812
Adjacent sequences: A005106 A005107 A005108 * A005110 A005111 A005112
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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EXTENSIONS
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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
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STATUS
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approved
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