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 A005109 Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1. (Formerly M0673) 54
 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 (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 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 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) C. K. Caldwell, The Prime Pages 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. James Pierpont, On an Undemonstrated Theorem of the Disquisitiones Arithmeticae, American Mathematical Society Bulletin 2 (1895-1896) pp. 77-83. 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; 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 *) 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 (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 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 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 October 23 05:56 EDT 2019. Contains 328335 sequences. (Running on oeis4.)