[seqfan] Asymptotic for A074753
franktaw at netscape.net
franktaw at netscape.net
Tue Jan 12 18:22:26 CET 2010
Since my last asymptotic question got no response :-(, I thought I
would try another one :-).
For a(n) = http://www.research.att.com/~njas/sequences/A074753 (Number
of integers k such that sigma(k) < n), what is the limit n->infinity
a(n) / n?
Up to 1 million, it appears to be about .673.
Certainly the limit is less than 1; any 2k > 2n/3 will have sigma(2k) >
n, so the limit must be < 5/6. Similar arguments with other factors
can refine this; looking only at prime factors up to 100000, I get an
upper bound of .7044..., and this clearly converges to something in
this neighborhood. I don't see any way to prove the actual limit is >
0, however.
Franklin T. Adams-Watters
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