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%I M0673 #133 Mar 27 2024 08:10:21
%S 2,3,5,7,13,17,19,37,73,97,109,163,193,257,433,487,577,769,1153,1297,
%T 1459,2593,2917,3457,3889,10369,12289,17497,18433,39367,52489,65537,
%U 139969,147457,209953,331777,472393,629857,746497,786433,839809,995329,1179649,1492993,1769473,1990657
%N Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.
%C 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
%C See A005105 for the definition of class r+ primes.
%C 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.
%C Sequence gives primes solutions to p == +1 (mod phi(p-1)). - _Benoit Cloitre_, Feb 22 2002
%C 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
%C 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
%C The subset of A055600 which are prime. - _Robert G. Wilson v_, Jul 19 2014
%C Named after the American mathematician James Pierpont (1866-1938). - _Amiram Eldar_, Jun 09 2021
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, section A18, p. 66.
%D George E. Martin, Geometric Constructions, Springer, 1998. ISBN 0-387-98276-0.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert G. Wilson v, <a href="/A005109/b005109.txt">Table of n, a(n) for n = 1..8396</a> (terms 1..795 from T. D. Noe, terms 796..1602 from Joerg Arndt)
%H Claudi Alsina and Roger B. Nelson, <a href="https://bookstore.ams.org/view?ProductCode=DOL/58">A Panoply of Polygons</a>, Dolciani Math. Expeditions Vol. 58, AMS/MAA (2023), see page 112.
%H Chris K. Caldwell, <a href="http://primes.utm.edu">The Prime Pages</a>.
%H David A. Cox and Jerry Shurman, <a href="http://www.jstor.org/stable/30037571">Geometry and number theory on clovers</a>, Amer. Math. Monthly, Vol. 112, No. 8 (2005), pp. 682-704.
%H Andrew M. Gleason, <a href="http://www.jstor.org/stable/2323624">Angle Trisection, the Heptagon and the Triskaidecagon</a>, American Mathematical Monthly, Vol. 95, No. 3 (1988), pp. 185-194.
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H Joel C. Langer and David A. Singer, <a href="https://doi.org/10.1155/2013/897320">Subdividing the trefoil by origami</a>, Geometry, Vol. 2013 (Hindawi Publishing Company, 2013), Article ID 897320. - From _N. J. A. Sloane_, Feb 08 2013
%H James Pierpont, <a href="https://doi.org/10.1090/S0002-9904-1895-00317-1">On an Undemonstrated Theorem of the Disquisitiones Arithmeticae</a>, American Mathematical Society Bulletin, Vol. 2, No. 3 (1895-1896), pp. 77-83.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierpontPrime.html">Pierpont Prime</a>.
%H <a href="/index/Pri#primes_Erdos_Selfridge">Index entries for sequences related to the Erdos-Selfridge classification</a>
%F 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
%F {primes p: A126805(PrimePi(p)) = 1}. - _R. J. Mathar_, Sep 24 2012
%e 97 = 2^5*3 + 1 is a term.
%t 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 &]]
%t Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* _Ray Chandler_, Nov 01 2005 *)
%t 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 *)
%o (PARI)
%o N=10^8; default(primelimit,N);
%o pq(p)={p-=1; (p/(2^valuation(p,2)*3^valuation(p,3)))==1;}
%o forprime(p=2,N,if(pq(p),print1(p,", ")));
%o /* _Joerg Arndt_, Sep 22 2012 */
%o (PARI) /* much more efficient: */
%o A005109_upto(lim=1e10)={my(L=List(), k2=1);
%o until ( lim <= k2 *= 2, my(k23 = k2);
%o until ( lim <= k23 *= 3, isprime(k23+1) && listput(L, k23+1));
%o ); Set(L) } /* _Joerg Arndt_, Sep 22 2012, edited by _M. F. Hasler_, Mar 17 2024 */
%o (Magma) [p: p in PrimesUpTo(10^8) | forall{d: d in PrimeDivisors(p-1) | d le 3}]; // _Bruno Berselli_, Sep 24 2012
%o (PARI)
%o N=10^8; default(primelimit, N);
%o print1("2, 3, ");forprime(p=5,N,if(omega(p-1)==3-p%3,print1(p", "))) \\ _Chris Boyd_, Mar 22 2014
%o (GAP)
%o K:=10^7;; # to get all terms <= K.
%o A:=Filtered([1..K],IsPrime);;
%o B:=List(A,i->Factors(i-1));;
%o C:=[];; for i in B do if Elements(i)=[2] or Elements(i)=[2,3] then Add(C,Position(B,i)); fi; od;
%o A005109:=Concatenation([2],List(C,i->A[i])); # _Muniru A Asiru_, Sep 10 2017
%o (Python)
%o from itertools import islice
%o from sympy import nextprime
%o def A005109_gen(): # generator of terms
%o p = 2
%o while True:
%o q = p-1
%o q >>= (~q&q-1).bit_length()
%o a, b = divmod(q,3)
%o while not b:
%o a, b = divmod(q:=a,3)
%o if q==1:
%o yield p
%o p = nextprime(p)
%o A005109_list = list(islice(A005109_gen(),30)) # _Chai Wah Wu_, Mar 17 2023
%Y Cf. A048135, A048136, A056637, A077497, A077498, A077500, A081426, A122259, A019434, A000668, A000040, A003586.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Comments and additional references from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
%E More terms from _David W. Wilson_
%E More terms from _Benoit Cloitre_, Feb 22 2002
%E More terms from _Robert G. Wilson v_, Mar 20 2003
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