%I #5 Sep 17 2018 12:18:42
%S 1,1,0,3,-7,18,-70,144,-573,1723,-6890,13786,-82714,165432,-661724,
%T 2646901,-13234503,26469012,-158814070,317628146,-1905768872,
%U 7623075492,-30492301966,60984603940,-487876831517,1463630494555,-5854521978216,23418087912870,-140508527477218
%N a(1) = 1; a(n) = tau(n) - tau(n-1)* a(n-1) if n > 1.
%C 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.
%H Harvey P. Dale, <a href="/A079898/b079898.txt">Table of n, a(n) for n = 1..1000</a>
%t 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]
%t 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 *)
%K sign
%O 1,4
%A _Joseph L. Pe_, Feb 20 2003