

A003401


Numbers of edges of regular polygons constructible with ruler and compass.
(Formerly M0505)


20



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
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OFFSET

1,2


COMMENTS

The terms 1 and 2 correspond to degenerate polygons.
These are also the numbers for which phi(n) is a power of 2: A209229(A000010(a(n)) = 1.  Olivier Gérard Feb 15 1999
A004729 and A051916 are subsequences. [Reinhard Zumkeller, Mar 20 2010]


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.
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. 12, 1953, Vol. 1, p. 187.


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, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 460. Original (Latin)
Eric Weisstein's World of Mathematics, Constructible Polygon
Eric Weisstein's World of Mathematics, Regular Polygon
Eric Weisstein's World of Mathematics, Trigonometry
Eric Weisstein's World of Mathematics, Trigonometry Angles


FORMULA

Computable as numbers such that cototientoftotient equals the totientoftotient: Flatten[Position[Table[co[eu[n]]eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=meu[m].  Labos Elemer, Oct 19 2001
Any product of 2^k and distinct Fermat primes (primes of the form 2^(2^m)+1).  Sergio Pimentel, Apr 30 2004, edited by Franklin T. AdamsWatters, 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]


EXAMPLE

34 is a term of this series because a circle can be divided exactly in 34 parts. 7 is not.


MATHEMATICA

Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ] (* Olivier Gérard Feb 15 1999 *)
(* 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] (* Robert G. Wilson v, 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]]]
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 1300; Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}] (* Robert G. Wilson v, Jul 28 2014 *)


PROG

(Haskell)
a003401 n = a003401_list !! (n1)
a003401_list = map (+ 1) $ elemIndices 1 $ map a209229 a000010_list
 Reinhard Zumkeller, Jul 31 2012
(PARI) for(n=1, 10^4, my(t=eulerphi(n)); if(t/2^valuation(t, 2)==1, print1(n, ", "))); \\ Joerg Arndt, Jul 29 2014


CROSSREFS

Cf. A004169, A000215, A099884, A019434 (Fermat primes).
Sequence in context: A121492 A182418 A204580 * A242441 A064481 A067939
Adjacent sequences: A003398 A003399 A003400 * A003402 A003403 A003404


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, R. K. Guy


STATUS

approved



