A340922 a(n) is the position of phi(A038568(n)^2)/phi(A038569(n)^2) in the enumeration of the rationals by A038568 and A038569, where phi is A000010.

0, 1, 2, 19, 20, 3, 4, 35, 36, 9, 10, 239, 240, 55, 56, 57, 58, 13, 14, 83, 84, 16, 15, 1059, 1060, 255, 256, 23, 24, 259, 260, 265, 266, 25, 26, 615, 616, 145, 146, 39, 40, 272, 271, 1763, 1764, 423, 424, 427, 428, 435, 436, 51, 52, 443, 444, 947, 948, 241, 242

Offset: 0

Author

Hugo Pfoertner, Feb 19 2021

Keywords

• nonn

Examples

			  n                   0   1   2   3    4   5    6    7    8   9    10
  j/k                 1  1/2  2  1/3   3  2/3  3/2  1/4   4  3/4  4/3
  phi(j^2)/phi(k^2)   1  1/2  2  1/6   6  1/3   3   1/8   8  3/4  4/3
  a(n)                0   1   2   19  20   3    4    35  36   9    10
.
  n                  11    12   13    14   15    16    17   18   19   20
  j/k                1/5    5  2/5   5/2  3/5    5/3  4/5  5/4  1/6    6
  phi(j^2)/phi(k^2)  1/20  20  1/10   10  3/10  10/3  2/5  5/2  1/12  12
  a(n)               239  240   55    56   57    58    13   14   83   84
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Hongjian Li, Pingzhi Yuan, and Hairong Bai, Positive Rational Numbers of the Form phi(n^2)/phi(m^2), The American Mathematical Monthly, 128:2 (2021), 174-176.
• Rémy Sigrist, PARI program for A340922

Crossrefs

Cf. A000010, A038568, A038569.

Programs

Julia

using Nemo
function A340922List(len)
    num(a) = euler_phi(numerator(a)^2)
    den(a) = euler_phi(denominator(a)^2)
    a, q, A, R = QQ(0), QQ(0), [], Int[]
    for n in 1:len
        q = next_minimal(q)
        x = num(q)//den(q)
        while true
            i = findfirst(isequal(x), A)
            if i == nothing
                a = next_minimal(a)
                push!(A, a)
            else
                push!(R, i - 1)
                break
            end
        end
    end
    R
end
A340922List(59) |> println # Peter Luschny, Feb 19 2021

PARI

\\ It is assumed that a38568 and a38569 are available as vectors,
\\ e.g. from the corresponding b-files.
\\ a38568=readvec("[path] a38568"); a38569=readvec("[path] a38569");
findinlist(n,d)={my(num=numerator(n/d),den=denominator(n/d));for(k=1,#a38568,if(num==a38568[k]&&den==a38569[k],return(k)));0};
for(k=1,60,my(m=a38568[k],n=a38569[k],num=eulerphi(m^2),den=eulerphi(n^2));print1(findinlist(num,den)-1,", "))