Question about A018804
Matthew Vandermast
ghodges14 at msn.com
Wed Dec 3 01:05:17 CET 2003
On 1 December, Leroy Quet wrote:
I figured (and Martin Cohen confirmed {referencing Hardy & Wright theorem
320}) that d(m), the number of positive divisors of m, was such that
limit{n-> infinity} (1/n) (sum{m=1 to n} d(m)) - ln(n)
= 2*c - 1, where c is Euler's constant (.5772...).
*******
Does anyone know, and if so could someone please tell me, whether the "typical" nth term of sequence A018804, the sum of gcd (k,n) for 1 <= k <= n
(a (1) through a (10) are 1, 3, 5, 8, 9, 15, 13, 20, 21, 27 . . . ; see http://www.research.att.com/projects/OEIS?Anum=A018804),
is about the same size as the nth term of sequence A006218, sum_(k=1 . . n) d (k ) (corresponding terms are 1, 3, 5, 8, 10, 14, 16, 20, 23, 27 . . . ; see http://www.research.att.com/projects/OEIS?Anum=A006218),
or "about" n* (ln (n ) + 2*c - 1?
Put more technically, does
limit {n -> infinity} sum_(k=1 . . . n) A018804(k )/ sum_(k=1 . . . n) A006218(k )
=1 (or possibly slightly more or less than 1)?
I can see why this might be true (since A006218 is also sum_(k=1 . . . n) floor (n/k) ), but proving it is beyond me. Thanks in advance to anyone who might reply to this query.
Regards,
Matthew Vandermast
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