Search: id:A023022
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%I A023022
%S A023022 1,1,1,2,1,3,2,3,2,5,2,6,3,4,4,8,3,9,4,6,5,11,4,10,6,9,6,14,4,15,8,10,
               8,
%T A023022 12,6,18,9,12,8,20,6,21,10,12,11,23,8,21,10,16,12,26,9,20,12,18,14,29,
               8,
%U A023022 30,15,18,16,24,10,33,16,22,12,35,12,36,18,20,18,30,12,39,16,27,20,41,
               12
%N A023022 Number of partitions of n into 2 ordered relatively prime parts. After 
               initial term, this is the "half-totient" function phi(n)/2.
%C A023022 The number of distinct linear fractional transformations of order n. 
               Also the half-totient function can be used to construct a tree containing 
               all the integers. On the zeroth rank we have just the integers 1 
               and 2 : immediate "ancestors" of 1 and 2 are (1: 3,4,6 2: 5,8,10,
               12) etc. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 03 2002
%C A023022 Moebius transform of floor(n/2). - Paul Barry (pbarry(AT)wit.ie), Mar 
               20 2005
%C A023022 Also number of different kinds of regular n-gons, one convex, the others 
               self-intersecting. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 20 2005
%C A023022 Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 28 2008: (Start)
%C A023022 Degree of polynomial which one of the root is Cos[2Pi/n]. These polynomials 
               are:
%C A023022 1: x-1
%C A023022 2: x+1
%C A023022 3: x+1/2
%C A023022 4: x
%C A023022 5: x-1/4
%C A023022 6: -4 + 2 x + x^2
%C A023022 7: x-1/2
%C A023022 8: -1 - 4 x + 4 x^2 + 8 x^3
%C A023022 9: x^2 - 1/2, 1 - 6 x + 8 x^3
%C A023022 10: -1 - 2 x + 4 x^2
%C A023022 11: 1 + 6 x - 12 x^2 - 32 x^3 + 16 x^4 + 32 x^5
%C A023022 12: x^2 - 3/4
%C A023022 13: -1 + 6 x + 24 x^2 - 32 x^3 - 80 x^4 + 32 x^5 + 64 x^6
%C A023022 14: 1 - 4 x - 4 x^2 + 8 x^3
%C A023022 15: 1 + 8 x - 16 x^2 - 8 x^3 + 16 x^4
%C A023022 16: 1 - 8 x^2 + 8 x^4
%C A023022 17: 1 - 8 x - 40 x^2 + 80 x^3 + 240 x^4 - 192 x^5 - 448 x^6 + 128 x^7 
               + 256 x^8
%C A023022 18: -1 - 6 x + 8 x^3
%C A023022 19: 1 + 10 x - 40 x^2 - 160 x^3 + 240 x^4 + 672 x^5 - 448 x^6 - 1024 
               x^7 + 256 x^8 + 512 x^9
%C A023022 5 20: - 20 x^2 + 16 x^4
%C A023022 21: 1 - 16 x + 32 x^2 + 48 x^3 - 96 x^4 - 32 x^5 + 64 x^6
%C A023022 22: -1 + 6 x + 12 x^2 - 32 x^3 - 16 x^4 + 32 x^5
%C A023022 23: -1 - 12 x + 60 x^2 + 280 x^3 - 560 x^4 - 1792 x^5 + 1792 x^6 + 4608 
               x^7 - 2304 x^8 - 5120 x^9 + 1024 x^10 + 2048 x^11
%C A023022 24: 1 - 16 x^2 + 16 x^4
%C A023022 25: -1 + 10 x + 100 x^2 - 40 x^3 - 800 x^4 + 32 x^5 + 2240 x^6 - 2560 
               x^8 + 1024 x^10
%C A023022 26: -1 - 6 x + 24 x^2 + 32 x^3 - 80 x^4 - 32 x^5 + 64 x^6
%C A023022 27: 1 + 18 x - 240 x^3 + 864 x^5 - 1152 x^7 + 512 x^9
%C A023022 28: -7 + 56 x^2 - 112 x^4 + 64 x^6
%C A023022 29: -1 + 14 x + 112 x^2 - 448 x^3 - 2016 x^4 + 4032 x^5 + 13440 x^6 - 
               15360 x^7 - 42240 x^8 + 28160 x^9 + 67584 x^10 - 24576 x^11 - 53248 
               x^12 + 8192 x^13 + 16384 x^14
%C A023022 30: 1 - 8 x - 16 x^2 + 8 x^3 + 16 x^4
%C A023022 etc. All polynomials which one of the roots is Cos[2Pi/n] (for rational 
               n)
%C A023022 belonging to solvable Galois groups, what mean that are available to 
               express by radicals. (End)
%D A023022 G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 
               1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problems 60&61.
%H A023022 T. D. Noe, <a href="b023022.txt">Table of n, a(n) for n=2..10000</a>
%H A023022 K. S. Brown, <a href="http://www.mathpages.com/home/kmath168.htm">The 
               Half-Totient Tree</a>
%H A023022 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Sequences realized as Parker vectors ...</
               a>, J. Integer Seqs., Vol. 6, 2003.
%H A023022 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PolygonTrianglePicking.html">Polygon Triangle Picking</a>
%H A023022 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrigonometryAngles.html">Trigonometry Angles</a>
%F A023022 phi(n)/2 for n >= 3.
%F A023022 a(n) = Sum(k/n: 1<=k<n and GCD(n, k)=1) = A023896(n)/n for n>2. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2005
%t A023022 Table[ EulerPhi[n]/2, {n, 3, 50}]
%Y A023022 Cf. A000010, A055684, A046657, A049806, A049703, A062956.
%Y A023022 Sequence in context: A070804 A104481 A078709 this_sequence A100677 A083290 
               A121842
%Y A023022 Adjacent sequences: A023019 A023020 A023021 this_sequence A023023 A023024 
               A023025
%K A023022 nonn
%O A023022 2,4
%A A023022 David W. Wilson (davidwwilson(AT)comcast.net)

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