A156775 Number of iterations of x->(sigma(x)+phi(x))/2 until a non-integer or a previous term is reached, starting with x=n; a(n)=0 if this never happens.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 3, 2, 1, 4, 1, 3, 2, 4, 1, 3, 1, 1, 4, 3, 2, 1, 1, 4, 3, 2, 1, 9, 1, 3, 4, 2, 1, 7, 1, 1, 3, 2, 1, 8, 3, 2, 3, 2, 1, 4, 1, 8, 7, 1, 4, 3, 1, 2, 7, 6, 1, 1, 1, 4, 3, 4, 6, 5, 1, 2, 1, 2, 1, 5, 6, 5, 4, 3, 1, 9, 4, 3, 7, 6, 5, 4, 1, 1, 9, 1, 1, 5, 1, 9, 3

Offset: 1

Author

M. F. Hasler, Feb 15 2009

Keywords

• nonn

Comments

In [Guy 1997] the iteration is said to fracture when sigma(x)+phi(x) becomes odd. It is not known if a(n)=0 for some n.

A156776(n) gives the number of iterations until the sequence fractures, resp. 0 if this never happens.

Examples

			Let f(x)=(sigma(x)+phi(x))/2. For x=1 we have f(x) = (1+1)/2 = 1, i.e. after a(1)=1 iterations, the initial term 1 is encountered. For x=2 we have f(x) = (3+1)/2 = 2, so a(2)=1 for the same reason; idem for x=3 and x=5. For x=4 we have f(x) = (7+2)/2 = 9/2, the sequence "fractures" after a(4)=1 iterations. For x=6 we have f(x) = (12+2)/2 = 7, f(7) = (8+6)/2 = 7: after a(6)=2 iterations, there's a value already seen before.
		

Links

• Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Crossrefs

Cf. A156776, A065387(n) = A000203(n) + A000010(n).

Programs

Mathematica

f[n_] := If[IntegerQ[n], n, 0]; g[n_] := f[(DivisorSigma[1, n] + EulerPhi[n])/2]; a[n_] := Module[{s = NestWhileList[g, n, UnsameQ, All]}, Length[s] - If[s[[-1]] == 0, 2, 1]]; Array[a, 105] (* Amiram Eldar, Apr 01 2024 *)

PARI

A156775(n,u=[])={ until( denominator( n=(sigma(n)+eulerphi(n))/2)>1 || setsearch(u,n), u=setunion(u,Set(n)));#u }