[seqfan] Re: An equivalence for integer sequences (with more questions than answers)
franktaw at netscape.net
franktaw at netscape.net
Sat Feb 28 00:15:38 CET 2009
While interesting in its own right, this is almost completely
unrelated to the original suggestion.
A closer match is to look at the asymptotic behavior of the
divergent sum. For example, Sum_{k=1}^n 1/k is
log(n) + o(log(n)) -- more precisely, log(n) + gamma + O(1/n)
-- so it is in the equivalence class of 1 * log(n). (N.b., log
here is the natural logarithm function, sometimes written ln,
not the base 10 logarithm. This is the standard in the
mathematical literature and in the OEIS.)
The even numbers (A005843 (excluding 0)) and the odd
numbers (A005408) are both in the class 1/2 * log(n), as are
the evil and odious numbers (A001969 (excluding 0) and
A000069). The primes (A000040) are in the class
1 * log(log(n)) -- see
<http://mathworld.wolfram.com/HarmonicSeriesofPrimes.html>.
I started looking at this from the beginning: A000001, the
number of groups of order n. I immediately got into a
number of interesting questions. Clearly, the asymptotic
behavior is at least n/log(n), since a(p) = 1 for every prime p.
The density of the non-primes such that a(n) = 1 (A050384)
is not immediately obvious. I think it includes most of the
semiprimes, which would make it at least on the order of
log(log(n))/log(n).
This sequence (A050384) -- or equivalently A003277, which
includes the primes -- might have finite density, although as
a subsequence of the square-free numbers, it cannot have
density greater than 6/pi^2. If it20does have finite density,
then A000001 would be in the class of c*n for some c; the
question would then become, what is c?
Franklin T. Adams-Watters
-----Original Message-----
From: Paolo Lava <ppl at spl.at>
Neil and seqfans,
"More generally, for a strictly positive sequence with offset O,
we can assign the value Sum_{n >= O} 1/a(n).
If the sum doesn't converge, assign the value "NA" (Not Applicable)"
To avoid “NA” we could consider the terms of the sequence as
coefficients of a
simple continued fraction. For instance:
...
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