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A023022
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Number of partitions of n into 2 ordered relatively prime parts. After initial term, this is the "half-totient" function phi(n)/2.
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38
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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, 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, 30, 15, 18, 16, 24, 10, 33, 16, 22, 12, 35, 12, 36, 18, 20, 18, 30, 12, 39, 16, 27, 20, 41, 12
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,4
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COMMENTS
| 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
Moebius transform of floor(n/2). - Paul Barry (pbarry(AT)wit.ie), Mar 20 2005
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
Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 28 2008: (Start)
Degrees of minimal polynomials of cos(2*Pi/n). The first few are
1: x - 1
2: x + 1
3: 2*x + 1
4: x
5: 4*x^2 + 2*x - 1
6: 2*x - 1
7: 8*x^3 + 4*x^2 - 4*x - 1
8: 2*x^2 - 1
9: 8*x^3 - 6*x + 1
10: 4*x^2 - 2*x - 1
11: 32*x^5 + 16*x^4 - 32*x^3 - 12*x^2 + 6*x + 1
These polynomials have solvable Galois groups, so their roots can be expressed by radicals. (End)
a(n) is the number of rationals p/q in the interval [0,1] such that p + q = n. - Geoffrey Critzer, Oct 10 2011.
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REFERENCES
| G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problems 60&61.
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LINKS
| T. D. Noe, Table of n, a(n) for n=2..10000
K. S. Brown, The Half-Totient Tree
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
Eric Weisstein's World of Mathematics, Polygon Triangle Picking
Eric Weisstein's World of Mathematics, Trigonometry Angles
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FORMULA
| phi(n)/2 for n >= 3.
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
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MATHEMATICA
| Table[ EulerPhi[n]/2, {n, 3, 50}]
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PROG
| (PARI) a(n)=if(n<=2, 1, eulerphi(n)/2);
(PARI) /* for printing minimal polynomials of cos(2*Pi/n) */
default(realprecision, 110);
for(n=1, 33, print(n, ": ", algdep(cos(2*Pi/n), a(n))));
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CROSSREFS
| Cf. A000010, A055684, A046657, A049806, A049703, A062956.
Cf. A181875, A181876, A181877, A183918.
Sequence in context: A070804 A104481 A078709 * A177501 A100677 A188637
Adjacent sequences: A023019 A023020 A023021 * A023023 A023024 A023025
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| Polynomials edited by W. Lang, Jan 08 2011, with the consent of A. Jasinski.
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