A079898 a(1) = 1; a(n) = tau(n) - tau(n-1)* a(n-1) if n > 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

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

Sequence in context: A259885 A110578 A134045 * A173449 A270519 A361087

Adjacent sequences: A079895 A079896 A079897 * A079899 A079900 A079901

Keyword

sign

Author

Joseph L. Pe, Feb 20 2003

Status

approved