Highly Composite Numbers (last edit Aug1 2010)
(circa May 2006)
from OEIS A2182, it appears that the distance from any highly composite number (HCN)to the nearest prime will itself always be prime. The 1st 384 (see subsequent lower) have been checked. See the file link for the related csv file and C++ source file (OEIS A117825). My independent conjecture (turns out related to /same as Fortune's Conjecture) is the absolute offset will always be prime when >1
files:
http://billymac00.pbwiki.com/f/cap.cpp source (Borland compiler)
http://billymac00.pbwiki.com/f/cap.exe exe (Windows)
http://billymac00.pbwiki.com/f/cap.xls xls results 1-120
http://billymac00.pbwiki.com/f/cap2.xls xls results 120-172
http://billymac00.pbwiki.com/f/hcn385.txt text file of 1st 384+ offsets (thanks to mensanator@aol.com for this xtension)
http://billymac00.pbwiki.com/f/A117825_pari.txt Pari GP scripting
others questions: will "1" always be the most common entry in the sequence? must a prime always appear between HCNs? will every prime be encountered in the sequence (in approx order)? will every prime appear in the sequence ie none will be missed?
Mar 2008 got an email the other day claiming that my HCN conjecture (essentially Fortune's) holds thru at least the first 13000 HCN ....thanks to Charles Greathouse ...who also quickly found a flaw in the GC reformulation I just corrected
for comparison, here is a link to Fortune's conjecture:
http://planetmath.org/encyclopedia/FortunesConjecture.html
essentially Fortune conjectured that the distance from a primorial (A002110) UP to the next prime is itself prime
analyzing the Greathouse data and tossing null/0 nat log values, one derives the following fit for N vs a(N) for the OEIS sequence: lnY=0.81*log(N)-0.27
here is the OpenOffice file:
ln117825.ods. It shows the same trending as the Prime Counting Function. Thus, it appears that as the #total primes against position flattens out asymptotically, so do the (nontrivial) prime distances related to this conjecture...also, the mode for Greathouse's dataset is still = 1 ...
Feb 9 2009 I noticed that the entries of A002182 show a lot of overlap with P(n,m), permutations of n things taken m at a time. For example, A(37)=665280 which is P(12,6). I did not check beyond the first 43 or so.
A nice OEIS sequence related to HCN is A000705: http://oeis.org/classic/A000705 . It relates to A002201
links to my other math pages:
Twin Primes and Goldbach's Conjecture Primality, GIMPS, and other prime stuff main