# Number theory

Prompted by the Iranian election where the results were made up on a spreadsheet and violated Benford’s first digit law I was playing with a spreadsheet, and for some reason, decided to count the number of times a set of numbers was evenly divisible by other numbers. (This of course is not Benford’s law at all).
So taking, for example, the numbers from 1 to 1000, the column below “SERIES” indicates that in this set with 1000 entries, the elements are evenly divisible by two 500 times, by three 333 times, etc. which is intuitive.
Taking the cumulative of numbers from 1 to 1000 (so the series is 1, 3=1+2, 6=3+3, 10=6+4, 15=10+5 … 500500=499500+1000), we get the column “CUMULATIVES”. In this case, numbers in this series are evenly divisible by two 500 times, but now threes show up 666 times. Fours are about half of two, and eights are one-fourth of two, and fives are twice tens, but six is half threes, not a third of twos.
The column RATIO is just the ratio of the SERIES and CUMULATIVES numbers.
I am sure there a good intuitive explanation for this fact, and I am sure mathematicians would find it obvious, but it surprised me.
I have no clue if this has any deeper meaning.

 Evenly Divisible by SERIES CUMULATIVES RATIO twos 500 500 1 threes 333 666 2 fours 250 250 1 fives 200 400 2 sixes 166 333 2 sevens 142 285 2 eights 125 124 1 nines 111 222 2 tens 100 200 2 none of the above 228 72 0.31

(none of the above does not imply prime status, but prime numbers will be none of the above).
3/25/2010 – corrected cumulatives column as per J.Horwitz