Using your Head is Permitted

April 2008 riddle

Let us divide the prime numbers into bins. Prime p will go into bin number floor(p/1000).

Let us pick a subset of the naturals by first selecting a set of bins, then taking all the naturals that have only primes inside the selected bins as their factors. (So, for example, to pick the number 3999991=1997*2003, we will need to first select at least both of bin number 1 [for 1997] and bin number 2 [for 2003]).

The harmonic series is the sum of 1/n over n=1,2,...

This series is known to diverge.

By selecting a set of bins, we are effectively picking a subset of the naturals. Let us calculate the partial harmonic series that is calculated solely over this (infinite) subset.

This month's question: what is the minimum number of bins that needs to be selected for the sub-series to diverge?

Prove your answer.

List of solvers:

Hongcheng Zhu (1 April 13:11)
Rani Hod (2 April 01:18)
Omer Angel (2 April 02:03)
Ross Millikan (3 April 06:23)
Mark Tilford (5 April 02:48)
Itsik Horovitz (15 April 17:15)
Bojan Bašić (20 April 01:15)
Albert Stadler (21 April 23:17)
David Jager (29 April 00:11)

Elegant solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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