A347561 - OEIS

%I #30 Oct 24 2021 08:45:42

%S 4,11,13,19,47,719,1439,2879,4079,4127,5807,6047,7247,7727,9839,10799,

%T 11279,13967,14159,15647,21599,24527,28319,28607,42767,44687,45887,

%U 48479,51599,51839,67247,68639,72767,77279,79967,81647,84047,84719,89087

%N Numbers m such that Conv(b,m) = b has a unique nontrivial solution (b = 0 and b = 1 are considered trivial solutions). Here, Conv(b,m) denotes the limit of b^^t (mod m) as t goes to infinity.

%C A101793 is a subsequence.

%C It appears that a(n) = A101793(n-4) for n>=5.

%C Except for n = 1, a(n) is prime.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>

%H <a href="/index/Te#tetration">Index entries for sequences related to tetration</a>

%e For a(2), we have:

%e Conv(2,11) = 9

%e Conv(3,11) = 9

%e Conv(4,11) = 4

%e Conv(5,11) = 1

%e Conv(6,11) = 5

%e Conv(7,11) = 2

%e Conv(8,11) = 3

%e Conv(9,11) = 5

%e Conv(10,11) = 1

%e Therefore, the only solution is Conv(4,11) = 4.

%t Conv[b_,m_] :=

%t Which[

%t Mod[b,m]==0,Return[0],

%t Mod[b,m]==1,Return[1],

%t GCD[b,m]==1,Return[PowerMod[b,Conv[b,MultiplicativeOrder[b,m]],m]],

%t True,Return[PowerMod[b,EulerPhi[m]+Conv[b,EulerPhi[m]],m]]

%t ]

%t a[m_] := Count[Table[Conv[b,m]==b,{b,0,m-1}],True]

%t Table[If[a[i]==3,i,## &[]],{i,2,800}]

%o (PARI) conv(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^conv(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + conv(b, eulerphi(n))));}

%o isok(m) = sum(b=2, m-1, conv(b, m) == b) == 1; \\ _Michel Marcus_, Sep 13 2021

%Y Cf. A347560, A343073, A000040, A101793, A183613.

%K nonn

%O 1,1

%A _Bernat Pagès Vives_, Sep 06 2021