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a(1) = 1; a(n) = sigma(n) - sigma(n-1)* a(n-1) if n > 1.
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%I #3 Feb 11 2014 19:05:41

%S 1,2,-2,15,-99,606,-7264,58127,-871892,11334614,-204023040,2448276508,

%T -68551742210,959724390964,-23033385383112,552801249194719,

%U -17136838725036271,308463097050652917,-12030060784975463743,240601215699509274902,-10105251059379389545852

%N a(1) = 1; a(n) = sigma(n) - sigma(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 sigma(n). In general the integral oscillator of s(n) can be defined similarly.

%t 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]

%K sign

%O 1,2

%A _Joseph L. Pe_, Feb 20 2003