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

1,1

Comments

A101793 is a subsequence.

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

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

Links

Table of n, a(n) for n=1..39.

Wikipedia, Tetration

Index entries for sequences related to tetration

Example

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A154040 A066985 A228003 * A234903 A369615 A095797

Adjacent sequences: A347558 A347559 A347560 * A347562 A347563 A347564

Keyword

nonn

Author

Bernat Pagès Vives, Sep 06 2021

Status

approved