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a(n) is the number of solutions of x^2 = eulerphi(x * m) where x is A293928(n).
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%I #26 Jan 07 2025 09:49:11

%S 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 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 1

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

%C The valid values of m in the equation are the terms of the sequence A151999 in order.

%C m is a solution if all squarefree divisors of x also divide m.

%C 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 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 The smallest solution where solutions exist are the terms of the sequence A055744 not in order.

%C The values of phi(m) are the terms of the sequence A068997 not in order.

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI scripts for various problems</a>

%F 0 < (phi(m)^(k+1) = phi(phi(m)^k*m)), k >= 1, m >= 1.

%e The first 1 is a term since there is only 1 solution when phi(m)=6. The solution is m=18.

%e The first 5 is a term since there are 5 solutions when phi(m)=16. These are 32, 34, 40, 48, and 60.

%e From _Michel Marcus_, Nov 08 2017: (Start)

%e Illustration of first few terms:

%e 1: [1, 2],

%e 2: [4, 6],

%e 4: [8, 10, 12],

%e 6: [18],

%e 8: [16, 20, 24, 30],

%e 12: [36, 42],

%e 16: [32, 34, 40, 48, 60],

%e 18: [54],

%e 20: [50],

%e 24: [72, 78, 84, 90],

%e 32: [64, 68, 80, 96, 102, 120],

%e ... (End)

%o (PARI) isok(n) = {iv = invphi(n); if (#iv, return (sum(m=1, #iv, n^2 == eulerphi(n*iv[m])))); return (0);}

%o 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 Cf. A000010, A006511, A032447, A151999, A055744, A068997, A293928.

%K nonn,easy

%O 1,1

%A _Torlach Rush_, Nov 05 2017