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

Author

Joseph L. Pe, Feb 20 2003

Keywords

• sign

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.

Crossrefs

Cf. A069942.

Programs

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]