A340922 - OEIS

%I #20 Feb 20 2021 02:57:54

%S 0,1,2,19,20,3,4,35,36,9,10,239,240,55,56,57,58,13,14,83,84,16,15,

%T 1059,1060,255,256,23,24,259,260,265,266,25,26,615,616,145,146,39,40,

%U 272,271,1763,1764,423,424,427,428,435,436,51,52,443,444,947,948,241,242

%N 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.

%H Rémy Sigrist, <a href="/A340922/b340922.txt">Table of n, a(n) for n = 0..10000</a>

%H Hongjian Li, Pingzhi Yuan, and Hairong Bai, <a href="https://doi.org/10.1080/00029890.2021.1850142">Positive Rational Numbers of the Form phi(n^2)/phi(m^2)</a>, The American Mathematical Monthly, 128:2 (2021), 174-176.

%H Rémy Sigrist, <a href="/A340922/a340922.gp.txt">PARI program for A340922</a>

%e   n                   0   1   2   3    4   5    6    7    8   9    10

%e   j/k                 1  1/2  2  1/3   3  2/3  3/2  1/4   4  3/4  4/3

%e   phi(j^2)/phi(k^2)   1  1/2  2  1/6   6  1/3   3   1/8   8  3/4  4/3

%e   a(n)                0   1   2   19  20   3    4    35  36   9    10

%e .

%e   n                  11    12   13    14   15    16    17   18   19   20

%e   j/k                1/5    5  2/5   5/2  3/5    5/3  4/5  5/4  1/6    6

%e   phi(j^2)/phi(k^2)  1/20  20  1/10   10  3/10  10/3  2/5  5/2  1/12  12

%e   a(n)               239  240   55    56   57    58    13   14   83   84

%o (PARI) \\ It is assumed that a38568 and a38569 are available as vectors,

%o \\ e.g. from the corresponding b-files.

%o \\ a38568=readvec("[path] a38568"); a38569=readvec("[path] a38569");

%o 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};

%o for(k=1,60,my(m=a38568[k],n=a38569[k],num=eulerphi(m^2),den=eulerphi(n^2));print1(findinlist(num,den)-1,", "))

%o (Julia)

%o using Nemo

%o function A340922List(len)

%o     num(a) = euler_phi(numerator(a)^2)

%o     den(a) = euler_phi(denominator(a)^2)

%o     a, q, A, R = QQ(0), QQ(0), [], Int[]

%o     for n in 1:len

%o         q = next_minimal(q)

%o         x = num(q)//den(q)

%o         while true

%o             i = findfirst(isequal(x), A)

%o             if i == nothing

%o                 a = next_minimal(a)

%o                 push!(A, a)

%o             else

%o                 push!(R, i - 1)

%o                 break

%o             end

%o         end

%o     end

%o     R

%o end

%o A340922List(59) |> println # _Peter Luschny_, Feb 19 2021

%Y Cf. A000010, A038568, A038569.

%K nonn

%O 0,3

%A _Hugo Pfoertner_, Feb 19 2021