A079883 a(1) = 1; a(n) = prime(n) - prime(n-1)* a(n-1) if n > 1.

1, 1, 2, -3, 32, -339, 4424, -75189, 1428614, -32858093, 952884728, -29539426531, 1092958781688, -44811310049165, 1926886332114142, -90563657609364621, 4799873853296324972, -283192557344483173287, 17274745998013473570574, -1157407981866902729228387

Offset

1,3

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. For s(n) = prime(n), one has s'(n) = 1 - (p(n-1)/p(n)) s'(n-1) = [p(n) - p(n-1) s'(n-1)]/p(n). The numerator is the expression p(n) - p(n-1) s'(n-1), which motivates the definition of the above sequence a(n). a(n) is called the "integral oscillator" of prime(n). In general the integral oscillator of s(n) can be defined similarly.

Links

Table of n, a(n) for n=1..20.

Mathematica

t = {1}; gt = 1; For[i = 2, i <= 24, i++, gt = Prime[i] - Prime[i - 1] gt; t = Append[t, gt]]; t ListPlot[t, PlotJoined -> True]

Crossrefs

Cf. A069942.

Sequence in context: A041895 A371117 A277481 * A358041 A066269 A083785

Adjacent sequences: A079880 A079881 A079882 * A079884 A079885 A079886

Keyword

sign

Author

Joseph L. Pe, Feb 20 2003

Status

approved