

A023022


Number of partitions of n into two relatively prime parts. After initial term, this is the "halftotient" function phi(n)/2 (A000010(n)/2).
(Formerly N0058)


77



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
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OFFSET

2,4


COMMENTS

The number of distinct linear fractional transformations of order n. Also the halftotient 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 ngons, one convex, the others selfintersecting.  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 ngon 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 A232631A232633.
(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 HalfTotient Tree
Tianxin Cai, Zhongyan Shen, and Mengjun Hu, On the Parity of the Generalized Euler Function, Advances in Mathematics (China), 2013, 42(4): 505510.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899907.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
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
Canze Zhu and Qunying Liao, A recursion formula for the generalized Euler function phi_e(n), arXiv:2105.10870 [math.NT], 2021.


FORMULA

a(n) = phi(n)/2 for n >= 3.
a(n) = (1/n)*Sum_{k=1..n1, 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
a(n) = Sum_{dn} moebius(n/d)*floor(d/2).  Michel Marcus, May 25 2021


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) [1] 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



