log(m)'s close to integers (sum convergence)

Leroy Quet qqquet at mindspring.com
Mon Mar 17 01:55:13 CET 2003

I am afraid that this question might either be trivial or already 
answered, in a way, a little while back when this group was discussing 
this sequence.

Consider the sequence:

>ID Number: A080021
>Sequence:  2,3,7,20,148,403,1096,1097,2980,2981,8103,59874,162755,
>           442413,1202604,3269017,8886110,8886111,24154952,24154953,
>           65659969,178482301,3584912846,9744803446,26489122130,
>           72004899337,195729609428
>Name:      log(n) is closer to an integer than is log(m) for any m with
>              2<=m<n.
>Comments:  Every term is floor(e^k)+r for some integers k and r with k>=1 
>and -1
>              <= r <= 1.
>References Suggested by Leroy Quet (qqquet at mindspring.com), Jan 19 2003
>Example:   log(2) = 1-0.306..., log(3) = 1+0.0986..., log(7) = 2-0.0540...,
>              log(20) = 3-0.00426...
>See also:  Cf. A080022, A080023.
>Keywords:  nonn
>Offset:    0
>Author(s): Dean Hickerson (dean at math.ucdavis.edu), Jan 20 2003
>Extension: More terms from Don Reble (djr at nk.ca), Jan 20 2003

Is the sum of the reciprocals of all of the terms (in the above sequence) 

ie. Does sum{k=0 to oo} 1/a(k)  converge?

(There are many terms close to each other, so the sum may very well 

Leroy Quet

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