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 A003401 Numbers of edges of regular polygons constructible with ruler and compass. (Formerly M0505) 34
 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) 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. 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 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, 3-20. [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, Mohr-Mascheroni theorem Wikipedia, Poncelet-Steiner theorem R. G. Wilson, V, Letter to N. J. A. Sloane, Aug. 1993 FORMULA Terms from 3 onward are 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, 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. 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[]]; 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), A295298. 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). 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 STATUS approved

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Last modified September 17 17:27 EDT 2019. Contains 327136 sequences. (Running on oeis4.)