[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




More information about the SeqFan mailing list