cp's OEIS Frontend

A294618 a(n) is the number of solutions of x^2 = eulerphi(x * m) where x is A293928(n).

%I A294618 #31 Jul 21 2025 02:42:06
%S A294618 2,2,3,1,4,2,5,1,1,4,6,3,3,5,1,7,6,4,1,7,1,3,1,8,10,5,1,1,9,3,8,4,1,9,
%T A294618 1,13,1,7,4,3,1,12,5,14,1,7,1,1,2,10,2,18,1,1,1,9,9,3,1,5,1,14,7,22,3,
%U A294618 1
%N A294618 a(n) is the number of solutions of x^2 = eulerphi(x * m) where x is A293928(n).
%C A294618 The valid values of m in the equation are the terms of the sequence A151999 in order.
%C A294618 m is a solution if all squarefree divisors of x also divide m.
%C A294618 The formula is recursive. For example, taking A151999(68) we get the following: 11664=phi(108*324), 1259712=phi(11664*324), 136048896=phi(1259712*324), ...
%C A294618 If a solution exists then x^(k+1) = phi(x^k * m) for a fixed m, and the smallest value of k must be 1. This follows from a|b implies phi(a)|phi(b), and for k >= 1 a^(k-1)|a^k.
%C A294618 The smallest solution where solutions exist are the terms of the sequence A055744 not in order.
%C A294618 The values of phi(m) are the terms of the sequence A068997 not in order.
%H A294618 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI scripts for various problems</a>
%F A294618 0 < (phi(m)^(k+1) = phi(phi(m)^k*m)), k >= 1, m >= 1.
%e A294618 The first 1 is a term since there is only 1 solution when phi(m)=6. The solution is m=18.
%e A294618 The first 5 is a term since there are 5 solutions when phi(m)=16. These are 32, 34, 40, 48, and 60.
%e A294618 From _Michel Marcus_, Nov 08 2017: (Start)
%e A294618 Illustration of first few terms:
%e A294618    1: [1, 2],
%e A294618    2: [4, 6],
%e A294618    4: [8, 10, 12],
%e A294618    6: [18],
%e A294618    8: [16, 20, 24, 30],
%e A294618   12: [36, 42],
%e A294618   16: [32, 34, 40, 48, 60],
%e A294618   18: [54],
%e A294618   20: [50],
%e A294618   24: [72, 78, 84, 90],
%e A294618   32: [64, 68, 80, 96, 102, 120],
%e A294618   ... (End)
%o A294618 (PARI) isok(n) = {iv = invphi(n); if (#iv, return (sum(m=1, #iv, n^2 == eulerphi(n*iv[m])))); return (0);}
%o A294618 lista(nn) = {for (n=1, nn, if (v = isok(n), print1(v, ", ")););} \\ \\ using the invphi script by Max Alekseyev; _Michel Marcus_, Nov 07 2017
%Y A294618 Cf. A000010, A006511, A032447, A151999, A055744, A068997, A293928.
%K A294618 nonn,easy
%O A294618 1,1
%A A294618 _Torlach Rush_, Nov 05 2017