

A003401


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


33



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
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 PonceletSteiner 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 mgon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)
Subsequence of A295298.  Antti Karttunen, Nov 27 2017


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.
David Eisenbud and Brady Haran, Heptadecagon and Fermat Primes (the math bit), Numberphile video (2015)
Mauro Fiorentini, Construibili (numeri)
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 460. Original (Latin)
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 320. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
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, MohrMascheroni theorem
Wikipedia, PonceletSteiner theorem
R. G. Wilson, V, Letter to N. J. A. Sloane, Aug. 1993


FORMULA

Terms from 3 onward are 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, clarified by Antti Karttunen, Nov 27 2017
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
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 !! (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
(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), A295298.
Edge lengths of other constructible mgons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17), A272536 (20).
Positions of zeros in A293516 (apart from two initial 1's), positions of ones in A295660.
Cf. also A046528.
Sequence in context: A182418 A204580 A295298 * A281624 A242441 A064481
Adjacent sequences: A003398 A003399 A003400 * A003402 A003403 A003404


KEYWORD

nonn,nice


AUTHOR

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


STATUS

approved



