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
Keywords
Examples
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.
Links
Programs
-
Mathematica
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] -
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))));} a(n) = sum(b=0, n-1, conv(b, n) == b); \\ Michel Marcus, Sep 13 2021
Comments