A156776 Number of iterations of x->(sigma(x)+phi(x))/2 until a non-integer is reached when starting with x=n; a(n)=0 if this never happens.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 3, 2, 1, 0, 1, 0, 2, 0, 0, 0, 4, 1, 0, 0, 4, 0, 0, 0, 1, 4, 3, 2, 1, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 7, 1, 1, 0, 0, 0, 8, 3, 2, 0, 0, 0, 0, 0, 8, 7, 1, 0, 0, 0, 0, 7, 6, 0, 1, 0, 0, 0, 4, 6, 5, 0, 0, 1, 0, 0, 5, 6, 5, 4, 3, 0, 9, 0, 0, 7, 6, 5, 4, 0, 1, 9, 1, 0, 5, 0, 9, 3

Offset

1,12

Comments

In [Guy 1997] the iteration is said to fracture when sigma(x)+phi(x) becomes odd. For n with a(n)=0, A156775(n) gives the number of iterations until a previously seen term is encountered.

Links

Table of n, a(n) for n=1..105.

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

Example

Let f(x)=(sigma(x)+phi(x))/2. For x=1 we have f(x) = (1+1)/2 = 1, i.e. this is a fixed point and the sequence will never fraction, hence a(1)=0. The same happens for x=2, 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, a fixed point, so again a(6)=a(7)=0.

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]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; Array[a, 105] (* Amiram Eldar, Apr 01 2024 *)

Prog

(PARI) A156776(n, u=[])={ until( denominator( n=(sigma(n)+eulerphi(n))/2)>1 || setsearch(u, n), u=setunion(u, Set(n))); if( denominator(n)>1, #u) }

Crossrefs

Cf. A156775, A065387(n) = A000203(n) + A000010(n).

Sequence in context: A154752 A271868 A194354 * A292108 A352910 A362328

Adjacent sequences: A156773 A156774 A156775 * A156777 A156778 A156779

Keyword

nonn

Author

M. F. Hasler, Feb 15 2009

Status

approved