1,2

1. Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to 1/2 or diverges. 3. s'(n) = 1 - (s(n-1)/s(n)) s'(n-1) = [s(n) - s(n-1) s'(n-1)]/s(n). The numerator is the expression s(n) - s(n-1) s'(n-1), which motivates the definition of the above sequence a(n). a(n) is called the "integral oscillator" of sigma(n). In general the integral oscillator of s(n) can be defined similarly.

t = {1}; gt = 1; For[i = 2, i <= 30, i++, gt = DivisorSigma[1, i] - DivisorSigma[1, i - 1] gt; t = Append[t, gt]]; t ListPlot[t, PlotJoined -> True]

Adjacent sequences: A079894 A079895 A079896 this_sequence A079898 A079899 A079900

Sequence in context: A032038 A009773 A006929 this_sequence A097540 A112327 A093114

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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Feb 20 2003