[seqfan] Which sequences maximize sigma(a(n))/(a(n) log log a(n) ) ?

Georgi Guninski guninski at guninski.com
Wed Nov 2 15:48:07 CET 2011

By Robin's theorem and RH 

sigma(a(n))/(a(n) log log a(n)) [1]

is bounded by e^gamma ~ 1.78107241799020 for n large enough.

For a(n) = lcm(1 .. n) [1] appears generally increasing and in about 12 hours of
computation for n = 10^8 -1000 I got the value 1.78106297350 which is relatively close
to the bound.

Empirically a(n)=lcm( 1..n) * product(prime p | p<log(n) ) gives bigger values.

Working with pari's multiprecision numbers and enough precision is much faster
and confirms the integer results so far. For n=10^9 for a(n)=lcm(1..n) I get
~ 1.78106978038 and for the second case slightly better (note this might well be incorrect).

Other sequences that maximize [1]?

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