cp's OEIS Frontend

A297475 Numbers n such that phi(x) = n for more than one value of x, and the smallest such x divides the largest.

%I A297475 #52 Nov 24 2024 11:14:26
%S A297475 1,2,8,10,22,28,30,44,46,52,54,56,58,66,70,78,82,92,102,104,106,110,
%T A297475 116,126,128,130,136,138,140,148,150,164,166,172,178,190,196,198,204,
%U A297475 210,212,222,226,228,238,250,260,262,268,270,282,292,294,296,306,310,316,330,332,342,344,346,356,358,366,368,372
%N A297475 Numbers n such that phi(x) = n for more than one value of x, and the smallest such x divides the largest.
%C A297475 The larger endpoint is always twice the value of the smaller endpoint.
%C A297475 Conjecture 1: The number of solutions, excluding endpoints is always 0, or an odd number. (known to n = 2 * 10^5)
%C A297475 Conjecture 2: If both endpoints are divisible by 5, then the number of solutions (excluding terms of A007366) is of the form 4k + 1. (known to n = 2 * 10^5)
%C A297475 A007366 is contained in this sequence and the number of solutions, excluding endpoints is always 0.
%C A297475 Terms of this sequence are totients with a single odd totient inverse.
%H A297475 Michel Marcus, <a href="/A297475/b297475.txt">Table of n, a(n) for n = 1..2441</a>
%H A297475 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%F A297475 2 = max({phi^-1(n)}) / min({phi^-1(n)}).
%F A297475 0 = A006511(n) mod A002181(n).
%e A297475 2 is in the sequence because {phi^-1(2)} = {3,4,6}, and 2 = 6 / 3.
%e A297475 8 is in the sequence because {phi^-1(8)} = {15,...,30}, and 2 = 30 / 15.
%e A297475 10 is in the sequence because {phi^-1(10)} = {11,22}, and 2 = 22 / 11.
%t A297475 With[{nn = 67}, Take[#, nn] &@ Keys@ Select[KeySort@ PositionIndex@ Array[EulerPhi, nn^2], IntegerQ[#2/#1] & @@ {First@ #, Last@ #} &]] (* _Michael De Vlieger_, Dec 31 2017 *)
%o A297475 (PARI) isok(n) = my(vx = invphi(n)); (#vx > 1) && ((vecmax(vx) % vecmin(vx)) == 0); \\ _Michel Marcus_, Jul 18 2018
%Y A297475 Cf. A002181, A002202, A006511, A007366, A032447.
%K A297475 nonn
%O A297475 1,2
%A A297475 _Torlach Rush_, Dec 30 2017