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A003401 Numbers of edges of regular polygons constructible with ruler and compass.
(Formerly M0505)
27
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

From Stanislav Sykora, May 02 2016: (Start)

The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 (A000079) are a subset and so are the Fermat primes (A019434), which are the only odd prime members.

The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).

Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)

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.

David Eisenbud and Brady Haran, Heptadecagon and Fermat Primes (the math bit) - Numberphile (2015)

Mauro Fiorentini, Construibili (numeri)

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 Number

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

Wikipedia, Constructible number

Wikipedia, Regular polygon

Wikipedia, Mohr-Mascheroni theorem

Wikipedia, Poncelet-Steiner theorem

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

log a(n) >> sqrt(n); if there are finitely many Fermat primes, then log a(n) ~ k log n for some k. - Charles R Greathouse IV, Oct 23 2015

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

(Python)

from sympy import totient

A003401_list = [n for n in range(1, 10**4) if format(totient(n), 'b').count('1') == 1]

# Chai Wah Wu, Jan 12 2015

CROSSREFS

Cf. A000079, A004169, A000215, A099884, A019434 (Fermat primes).

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17), A272536 (20).

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 December 7 05:36 EST 2016. Contains 278841 sequences.