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.

4, 11, 13, 19, 47, 719, 1439, 2879, 4079, 4127, 5807, 6047, 7247, 7727, 9839, 10799, 11279, 13967, 14159, 15647, 21599, 24527, 28319, 28607, 42767, 44687, 45887, 48479, 51599, 51839, 67247, 68639, 72767, 77279, 79967, 81647, 84047, 84719, 89087

Offset: 1

Author

Bernat Pagès Vives, Sep 06 2021

Keywords

• nonn

Comments

A101793 is a subsequence.

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

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

Examples

			For a(2), we have:
Conv(2,11) = 9
Conv(3,11) = 9
Conv(4,11) = 4
Conv(5,11) = 1
Conv(6,11) = 5
Conv(7,11) = 2
Conv(8,11) = 3
Conv(9,11) = 5
Conv(10,11) = 1
Therefore, the only solution is Conv(4,11) = 4.
		

Links

• Wikipedia, Tetration
• Index entries for sequences related to tetration

Crossrefs

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

Programs

Mathematica

Conv[b_,m_] :=
Which[
Mod[b,m]==0,Return[0],
Mod[b,m]==1,Return[1],
GCD[b,m]==1,Return[PowerMod[b,Conv[b,MultiplicativeOrder[b,m]],m]],
True,Return[PowerMod[b,EulerPhi[m]+Conv[b,EulerPhi[m]],m]]
]
a[m_] := Count[Table[Conv[b,m]==b,{b,0,m-1}],True]
Table[If[a[i]==3,i,## &[]],{i,2,800}]

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))));}
isok(m) = sum(b=2, m-1, conv(b, m) == b) == 1; \\ Michel Marcus, Sep 13 2021