The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A023022 Number of partitions of n into two relatively prime parts. After initial term, this is the "half-totient" function phi(n)/2 (A000010(n)/2). (Formerly N0058) 58
 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; text; internal format)
 OFFSET 2,4 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, Jun 03 2002 Moebius transform of floor(n/2). - Paul Barry, Mar 20 2005 Also number of different kinds of regular n-gons, one convex, the others self-intersecting. - Reinhard Zumkeller, Aug 20 2005 From Artur Jasinski, 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 It appears that, for n > 2, a(n) = A023896(n)/n. Also, it appears that a record occurs at n > 2 in this sequence if and only if n is a prime.  For example, records occur at n=5, 7, 11, 13, 17, ..., all of which are prime. - John W. Layman, Mar 26 2012 From Wolfdieter Lang, Dec 19 2013: (Start) a(n) is the degree of the algebraic number of s(n)^2 = (2*sin(Pi/n))^2, starting at a(1)=1. s(n) = 2*sin(Pi/n) is the length ratio side/R for a regular n-gon inscribed in a circle of radius R (in some length units). For the coefficient table of the minimal polynomials of s(n)^2 see A232633. Because for even n, s(n)^2 lives in the algebraic number field Q(rho(n/2)), with rho(k) = 2*cos(Pi/k), the degree is a(2*l) = A055034(l). For odd n, s(n)^2 is an integer in Q(rho(n)), and the degree is a(2*l+1) = A055034(2*l+1) = phi(2*l+1)/2, l >= 1, with Euler's totient phi=A000010 and a(1)=1. See also A232631-A232633. (End) Also for n > 2: number of fractions A182972(k)/A182973(k) such that A182972(k) + A182973(k) = n, A182972(n) and A182973(n) provide an enumeration of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator. - Reinhard Zumkeller, Jul 30 2014 Number of distinct rectangles with relatively prime length and width such that L + W = n, W <= L. For a(17)=8; the rectangles are 1 X 16, 2 X 15, 3 X 14, 4 X 13, 5 X 12, 6 X 11, 7 X 10, 8 X 9. - Wesley Ivan Hurt, Nov 12 2017 After including a(1) = 1, the number of elements of any reduced residue system mod* n used by Brändli and Beyne is a(n). See the examples below. - Wolfdieter Lang, Apr 22 2020 REFERENCES G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problems 60&61. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 2..10000 Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016. K. S. Brown, The Half-Totient Tree Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003. Pinthira Tangsupphathawat, Takao Komatsu, Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5. Eric Weisstein's World of Mathematics, Polygon Triangle Picking Eric Weisstein's World of Mathematics, Trigonometry Angles FORMULA a(n) = phi(n)/2 for n >= 3. a(n) = (1/n)*Sum_{k=1..n-1, gcd(n, k)=1} k = A023896(n)/n for n>2. - Reinhard Zumkeller, Aug 20 2005 G.f.: x*(x - 1)/2 + (1/2)*Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Apr 13 2017 EXAMPLE a(15)=4 because there are 4 partitions of 15 into two parts that are relatively prime: 14 + 1, 13 + 2, 11 + 4, 8 + 7. - Geoffrey Critzer, Jan 25 2015 The smallest nonnegative reduced residue system mod*(n) for n = 1 is {0}, hence a(1) = 1; for n = 9 it is {1, 2, 4}, because 5 == 4 (mod* 9) since -5 == 4 (mod 9), 7 == 2 (mod* 9) and 8 == 1 (mod* 9). Hence a(9) = phi(9)/2 = 3. See the comment on Brändli and Beyne above. - Wolfdieter Lang, Apr 22 2020 MAPLE A023022 := proc(n)     if n =2 then         1;     else         numtheory[phi](n)/2 ;     end if; end proc: seq(A023022(n), n=2..60) ; # R. J. Mathar, Sep 19 2017 MATHEMATICA Join[{1}, Table[EulerPhi[n]/2, {n, 3, 100}]] (adapted by Vincenzo Librandi, Aug 19 2018 ) PROG (PARI) a(n)=if(n<=2, 1, eulerphi(n)/2); /* 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)))); (Haskell) a023022 n = length [(u, v) | u <- [1 .. div n 2],                              let v = n - u, gcd u v == 1] -- Reinhard Zumkeller, Jul 30 2014 (Python) from sympy.ntheory import totient def a(n): return 1 if n<3 else totient(n)/2 # Indranil Ghosh, Mar 30 2017 (MAGMA)  cat [EulerPhi(n)/ 2: n in [3..100]]; // Vincenzo Librandi, Aug 19 2018 CROSSREFS Cf. A000010, A055684, A046657, A049806, A049703, A062956. Cf. A181875, A181876, A181877, A183918. Cf. A023896. Cf. A182972, A182973, A245497, A245718. Sequence in context: A253630 A104481 A078709 * A177501 A330926 A323273 Adjacent sequences:  A023019 A023020 A023021 * A023023 A023024 A023025 KEYWORD nonn,easy AUTHOR N. J. A. Sloane. This was in the 1973 "Handbook", but then was dropped from the database. Resubmitted by David W. Wilson. EXTENSIONS Entry revised by N. J. A. Sloane, Jun 10 2012 Polynomials edited with the consent of Artur Jasinski by Wolfdieter Lang, Jan 08 2011 Name clarified by Geoffrey Critzer, Jan 25 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 29 20:22 EDT 2020. Contains 337432 sequences. (Running on oeis4.)