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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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47
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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 (benoit7848c(AT)orange.fr), Feb 22 2002
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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.
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
| T. D. Noe, Pierpont primes less than 10^100; table of n, a(n) for n = 1..795
Eric Weisstein's World of Mathematics, Pierpont Prime
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FORMULA
| A122257(a(n)) = 1; A122258(n) = number of Pierpont primes <= n; A122260 gives numbers having only Pierpont primes as factors. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 29 2006
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EXAMPLE
| 97 = 2^5*3 + 1 is a member.
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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 (rayjchandler(AT)sbcglobal.net), Nov 01 2005)
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CROSSREFS
| Cf. A048135, A048136, A056637, A005105, A005110, A005111, A005112, A077497, A077498, A077500, A081424, A081425, A081426, A081427, A081428, A081429, A081430, A122259.
Sequence in context: A109461 A138539 A090422 * A179336 A080608 A137812
Adjacent sequences: A005106 A005107 A005108 * A005110 A005111 A005112
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KEYWORD
| nonn,nice,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Comments and additional references from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr). More terms from David W. Wilson (davidwwilson(AT)comcast.net)
More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 22 2002
More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 20 2003
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