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

0,3

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

Example

  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

Prog

(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, ", "))
(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

Crossrefs

Cf. A000010, A038568, A038569.

Sequence in context: A229264 A132129 A125611 * A370460 A022119 A042247

Adjacent sequences: A340919 A340920 A340921 * A340923 A340924 A340925

Keyword

nonn

Author

Hugo Pfoertner, Feb 19 2021

Status

approved