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A079898 a(1) = 1; a(n) = tau(n) - tau(n-1)* a(n-1) if n > 1. 1
1, 1, 0, 3, -7, 18, -70, 144, -573, 1723, -6890, 13786, -82714, 165432, -661724, 2646901, -13234503, 26469012, -158814070, 317628146, -1905768872, 7623075492, -30492301966, 60984603940, -487876831517, 1463630494555, -5854521978216, 23418087912870, -140508527477218 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A259885 A110578 A134045 * A173449 A270519 A361087
KEYWORD
sign
AUTHOR
Joseph L. Pe, Feb 20 2003
STATUS
approved

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Last modified June 24 22:33 EDT 2024. Contains 373690 sequences. (Running on oeis4.)