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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A022119 A042247 A041451

Adjacent sequences:  A340919 A340920 A340921 * A340923 A340924 A340925

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Feb 19 2021

STATUS

approved

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Last modified January 22 21:47 EST 2022. Contains 350504 sequences. (Running on oeis4.)