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
Keywords
Examples
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.
Links
- Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Programs
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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 *) -
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) }
Comments