A343073 -id:A343073 - cp's OEIS frontend

A347560 a(n) is the number of solutions to Conv(b,n)=b where Conv(b,n) denotes the limit of b^^t (mod n) as t goes to infinity.

2, 2, 3, 2, 5, 4, 5, 4, 5, 3, 8, 3, 7, 8, 7, 4, 9, 3, 10, 8, 8, 5, 14, 6, 9, 4, 12, 6, 15, 9, 7, 10, 9, 10, 14, 4, 9, 10, 18, 7, 19, 5, 13, 14, 10, 3, 20, 10, 13, 12, 14, 7, 9, 12, 20, 10, 13, 7, 28, 9, 15, 21, 11, 17, 24, 10, 14, 13, 22, 15, 24, 7, 9, 17, 17, 20, 24, 10, 28

Offset: 2

Bernat Pagès Vives, Sep 06 2021

nonn

Writing n = m^(2k), a(n) >= 2^A001221(n) + m^k - 1.
Writing n = m^(2k+1), a(n) >= 2^A001221(n) + m^k - 1.
If n is in A101793, then a(n) = 3.
It appears that a(n) = 2 only for n = 2, 3, 5.
It appears that a(n) = 3 only for n = 4, 11, 13, 19 and for n in A101793.
It is not known whether there exist infinitely many numbers n satisfying a(n) = 3.

			For n = 100, pick b = 3.
3^^1 ==  3 (mod 100)
3^^2 == 27 (mod 100)
3^^3 == 87 (mod 100)
3^^4 == 87 (mod 100)
3^^5 == 87 (mod 100)
...
It can be proved that the sequence converges to 87, so Conv(3,100) = 87. Since b = 3 does not satisfy Conv(b,100) = b, this value is not counted in a(100).
For n = 7, pick b = 2.
2^^1 == 2 (mod 7)
2^^2 == 4 (mod 7)
2^^3 == 2 (mod 7)
2^^4 == 2 (mod 7)
2^^5 == 2 (mod 7)
...
It can be proved that the sequence converges to 2, so Conv(2,7) = 2. Thus, 2 is a solution for a(7). The other 3 solutions are 0, 1 and 4 giving a total of a(7) = 4 solutions.

Wikipedia, Tetration
Index entries for sequences related to tetration

Cf. A347561, A343073, A000040, A101793, A183613, A001221.

Conv[b_,n_] :=
Which[
Mod[b,n]==0,Return[0],
Mod[b,n]==1,Return[1],
GCD[b,n]==1,Return[PowerMod[b,Conv[b,MultiplicativeOrder[b,n]],n]],
True,Return[PowerMod[b,EulerPhi[n]+Conv[b,EulerPhi[n]],n]]
]
a[n_] := Count[Table[Conv[b,n]==b,{b,0,n-1}],True]

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))));}
a(n) = sum(b=0, n-1, conv(b, n) == b); \\ Michel Marcus, Sep 13 2021

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.

Original entry on oeis.org

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

Bernat Pagès Vives, Sep 06 2021

nonn

A101793 is a subsequence.
It appears that a(n) = A101793(n-4) for n>=5.
Except for n = 1, a(n) is prime.

			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.

Wikipedia, Tetration
Index entries for sequences related to tetration

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

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}]

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

cp's OEIS Frontend

A347560 a(n) is the number of solutions to Conv(b,n)=b where Conv(b,n) denotes the limit of b^^t (mod n) as t goes to infinity.

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