OFFSET
1,4
COMMENTS
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 tau(n). In general the integral oscillator of s(n) can be defined similarly.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
MATHEMATICA
t = {1}; gt = 1; For[i = 2, i <= 30, i++, gt = DivisorSigma[0, i] - DivisorSigma[0, i - 1] gt; t = Append[t, gt]]; t ListPlot[t, PlotJoined -> True]
RecurrenceTable[{a[1]==1, a[n]==DivisorSigma[0, n]-DivisorSigma[0, n-1]a[n-1]}, a, {n, 30}] (* Harvey P. Dale, Sep 17 2018 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Joseph L. Pe, Feb 20 2003
STATUS
approved