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A003401
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Numbers of edges of polygons constructible with ruler and compass.
(Formerly M0505)
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19
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1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The terms 1 and 2 correspond to degenerate polygons.
These are also the numbers for which phi(n) is a power of 2.
A004729 and A051916 are subsequences. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 20 2010]
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REFERENCES
| A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
Allan Clark, Elements of Abstract Algebra, Chapter 4, Galois Theory, Dover Publications, NY 1984, page 124.
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 460.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..2000
T. Chomette, Construction des polygones reguliers
Bruce Director, Measurement and Divisibility.
C. F. Gauss, Disquisitiones Arithmeticae, Lipsiae, 1801. Reprinted in C. F. Gauss, Werke, 1863.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Trigonometry Angles
Eric Weisstein's World of Mathematics, Constructible Polygon
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FORMULA
| Computable as numbers such that cototient-of-totient equals the totient-of-totient: Flatten[Position[Table[co[eu[n]]-eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=m-eu[m]. - Labos E. (labos(AT)ana.sote.hu), Oct 19 2001
Any product of 2^k and distinct Fermat primes (primes of the form 2^(2^m)+1). - Sergio Pimentel (ferdiego(AT)cox-internet.com), Apr 30 2004, edited by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 16 2006
If the well known conjecture that, there are only five prime Fermat numbers F_k=2^{2^k}+1, k=0,1,2,3,4, then we have exactly: sum_{n=1,...,infty} 1/a(n)= 2*prod_{k=0,...,4} (1+1/F_k) = 4869735552/1431655765 = 3.40147098978.... [Vladimir Shevelev and T. D. Noe, Dec 01 2010]
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EXAMPLE
| 34 is a term of this series because a circle can be divided exactly in 34 parts. 7 is not.
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MATHEMATICA
| Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ]
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Take[ Union[ Flatten[ NestList[2# &, Times @@@ Table[ UnrankSubset[n, Join[{1}, Table[2^2^i + 1, {i, 0, 4}]]], {n, 63}], 11]]], 60] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 11 2005)
nn=10; logs=Log[2, {2, 3, 5, 17, 257, 65537}]; lim2=Floor[nn/logs[[1]]]; Sort[Reap[Do[z={i, j, k, l, m, n}.logs; If[z<=nn, Sow[2^z]], {i, 0, lim2}, {j, 0, 1}, {k, 0, 1}, {l, 0, 1}, {m, 0, 1}, {n, 0, 1}]][[2, 1]]]
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CROSSREFS
| Cf. A004169.
Cf. A000215, A099884.
Cf. A019434 (Fermat primes)
Sequence in context: A029032 A059809 A121492 * A064481 A067939 A067784
Adjacent sequences: A003398 A003399 A003400 * A003402 A003403 A003404
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
| Comment and program from Olivier Gerard (02/99).
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