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A156775
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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.
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2
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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
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OFFSET
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1,6
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) A156775(n, u=[])={ until( denominator( n=(sigma(n)+eulerphi(n))/2)>1 || setsearch(u, n), u=setunion(u, Set(n))); #u }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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