OFFSET
1,3
COMMENTS
Given the quasi-order and coach theorems of Hilton and Pedersen (cf. A003558): phi(b) = 2 * k * c with b odd. Dividing through by 2 we obtain phi(b)/2 = k * c which is the same as a(n) = A003558(n-1) * A135303(n-1). Here k refers to the "least exponent" (cf. A003558), while c is the number of coaches for odd b (cf. A135303). - Gary W. Adamson, Sep 25 2012
After the initial term, this is the totient function phi(2n-1)/2 (A037225(n-1)/2). Also a(n) is the number of partitions of the odd number (2n+1) into two ordered relatively prime parts. If p and q are such parts where p > q and p+q = 2n+1 then they can generate primitive Pythagorean triples of the form (p^2 - q^2, 2*p*q, p^2 + q^2). - Frank M Jackson, Oct 30 2012
REFERENCES
Peter Hilton and Jean Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, p. 200.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
FORMULA
a(1) = 1, and for n > 1, a(n) = phi(2n-1)/2. - Benoit Cloitre, Oct 11 2002
It would appear that a(n) = Sum_{k=0..n} abs(Jacobi(k, 2n-2k+1)). - Paul Barry, Jul 20 2005
a(n) = A055034(2*n-1), n >= 1. - Wolfdieter Lang, Feb 07 2020
G.f.: (x + Sum_{n>=1} mu(2n-1) * x^n * (1 + x^(2n-1)) / (1 - x^(2n-1))^2) / 2. - Mamuka Jibladze, Dec 13 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Feb 11 2023
EXAMPLE
MAPLE
A072451 := n -> ceil(numtheory:-phi(2*n-1)/2):
seq(A072451(n), n=1..73); # Peter Luschny, Feb 24 2020
MATHEMATICA
gw[x_] := Table[GCD[x, w], {w, 1, x}] rrs[x_] := Flatten[Position[gw[x], 1]] Table[Count[OddQ[rrs[2*w-1]], True], {w, 1, 128}]
(* Additional programs: *)
Table[Count[Range[1, #, 2], k_ /; CoprimeQ[k, #]] &[2 n - 1], {n, 73}] (* or *) Array[If[# == 1, #, EulerPhi[2 # - 1]/2] &, 73] (* Michael De Vlieger, Jul 24 2017 *)
PROG
(PARI) A072451(n) = if(1==n, n, eulerphi(n+n-1)/2); \\ (After Benoit Cloitre's formula) - Antti Karttunen, Jul 24 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jun 19 2002
STATUS
approved