A176839 The number of iterations to reach 1 under the map x -> x-tau(phi(x)), starting at n.

0, 1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 5, 5, 4, 6, 5, 6, 5, 7, 5, 7, 6, 7, 6, 8, 6, 7, 7, 7, 7, 9, 8, 8, 7, 8, 8, 10, 8, 9, 9, 11, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 10, 11, 11, 12, 11, 12, 12, 12, 12, 12, 12, 13, 11, 12, 13, 13, 12, 13, 13, 13, 12, 13, 14, 14, 14

Offset: 1

Author

Michel Lagneau, Apr 27 2010

Keywords

• nonn

Comments

Tau(n) = A000005(n) is the number of divisors of n, and phi(n) = A000010(n) is the Euler totient function.

Examples

			a(12)=4 because
f(12) = 12 - tau(phi(12)) = 12 - tau(4) = 12 - 3 = 9;
f(9) = 9 - tau(phi(9)) = 9 - tau(6) = 9 - 4 = 5;
f(5) = 5 - tau(phi(5)) = 5 - tau(4) = 5 - 3 = 2;
f(2) = 2 - tau(phi(2)) = 2 - tau(1) = 2 - 1 = 1, and a(12) = 4.
		

Crossrefs

Cf. A062821.

Programs

Maple

A062821 := proc(n)
        numtheory[tau](numtheory[phi](n)) ;
end proc:
A176839 := proc(n)
        a := 0 ;
        x := n ;
        while x <> 1 do
                x := x-A062821(x) ;
                a := a+1 ;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 11 2011

Mathematica

f[n_] := If[n == 1, 1, n - DivisorSigma[0, EulerPhi[n]]];
a[n_] := Length[FixedPointList[f, n]] - 2;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 09 2024 *)