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A347561 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.

%I A347561 #30 Oct 24 2021 08:45:42
%S A347561 4,11,13,19,47,719,1439,2879,4079,4127,5807,6047,7247,7727,9839,10799,
%T A347561 11279,13967,14159,15647,21599,24527,28319,28607,42767,44687,45887,
%U A347561 48479,51599,51839,67247,68639,72767,77279,79967,81647,84047,84719,89087
%N A347561 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 A347561 A101793 is a subsequence.
%C A347561 It appears that a(n) = A101793(n-4) for n>=5.
%C A347561 Except for n = 1, a(n) is prime.
%H A347561 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%H A347561 <a href="/index/Te#tetration">Index entries for sequences related to tetration</a>
%e A347561 For a(2), we have:
%e A347561 Conv(2,11) = 9
%e A347561 Conv(3,11) = 9
%e A347561 Conv(4,11) = 4
%e A347561 Conv(5,11) = 1
%e A347561 Conv(6,11) = 5
%e A347561 Conv(7,11) = 2
%e A347561 Conv(8,11) = 3
%e A347561 Conv(9,11) = 5
%e A347561 Conv(10,11) = 1
%e A347561 Therefore, the only solution is Conv(4,11) = 4.
%t A347561 Conv[b_,m_] :=
%t A347561 Which[
%t A347561 Mod[b,m]==0,Return[0],
%t A347561 Mod[b,m]==1,Return[1],
%t A347561 GCD[b,m]==1,Return[PowerMod[b,Conv[b,MultiplicativeOrder[b,m]],m]],
%t A347561 True,Return[PowerMod[b,EulerPhi[m]+Conv[b,EulerPhi[m]],m]]
%t A347561 ]
%t A347561 a[m_] := Count[Table[Conv[b,m]==b,{b,0,m-1}],True]
%t A347561 Table[If[a[i]==3,i,## &[]],{i,2,800}]
%o A347561 (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 A347561 isok(m) = sum(b=2, m-1, conv(b, m) == b) == 1; \\ _Michel Marcus_, Sep 13 2021
%Y A347561 Cf. A347560, A343073, A000040, A101793, A183613.
%K A347561 nonn
%O A347561 1,1
%A A347561 _Bernat Pagès Vives_, Sep 06 2021