[seqfan] An equivalence for integer sequences (with more questions than answers)

Jaume Oliver i Lafont joliverlafont at gmail.com
Thu Feb 26 18:40:14 CET 2009

Two integer sequences a(n) and b(n) are equivalent iff

Sum_{n=0..inf}1/a(n) = Sum_{n=0..inf}1/b(n)

In this equivalence relation, classes can be labeled with real numbers.
For example, the powers of two and Sylvester sequence belong to
equivalence class "1".
A001466 and A156618 are also equivalent in this sense and belong to
class "Pi-3".

We might classify the (zero-offset) sequence 1,-1,2,-2,3,-3... in
class 0, as well as -1,1,-2,2,-3,3...

a(n)=n+1 and b(n)=2(n+1) would probably fit into class +infinity, while
a(n)=-(n+1) and b(n)=-2(n+1) would belong to -infinity.

But where does a(n)=n go? And a(n)=-n? Do these go to different sign
infinities because all non-zero terms share the same sign or having
one zero is so strong that a different infinity is needed?
How about a(n)=0, zero everywhere?
Would it make any difference summing from -infinity to +infinity?

Sorry if too speculative or trivial...


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