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A003401 Numbers of edges of regular polygons constructible with ruler and compass.
(Formerly M0505)
19
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; text; internal format)
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. 1-2, 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 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 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. Adams-Watters, 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 !! (n-1)

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

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Last modified August 20 06:03 EDT 2014. Contains 245796 sequences.