report

[N. J. A. Sloane]

Dear SeqFans: Here are some announcements:

o I will be away from email Feb 25 - Mar 07

o Jeff Shallit is taking over as Editor-in-Chief of the 
     J. Integer Sequences. (My math typist Sue Pope was laid off
     and without her I couldn't run it.  Anyway, 4 years was enough.
     There were 99 submissions and 50 acceptances.)  I'm very
     glad Jeff is taking over. The URL will stay the same for now
     but will soon change to a Waterloo address.

o Frank Ellermann kindly identified a large number of duplicate
     sequences which have now been removed from the database

o When extending, editing or correcting a sequence please
     check that it is not already in the database.  By now
     all the obvious sequences are there!

o If you format sequences yourself please use 3-letter months
     and 2-digit dates, as in Jan 01, 2002
     (NOT February 4, 2002)

o The best recent sequence:

%I A066720
%S A066720 1,2,3,5,7,8,11,13,17,18,19,23,29,31,37,41,43,47,50,53,59,60,61,67,71,
%T A066720 73,79,81,83,89,97,98,101,103,105,107,109,113,127,128,131,137,139,149,
%U A066720 151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239
%N A066720 The greedy rational packing sequence: a(1) = 1; for n > 1, a(n) is 
smallest number such that the ratios a(i)/a(j) for 1 <= i < j <= n are all distinct.
%C A066720 If you replace the word "ratio" with "difference" and start from 1 using the 
same greedy algorithm you get A005282. - Sharon Sela (sharonsela at hotmail.com), Jan 15, 2002
%C A066720 Does every rational number appear as a ratio? See A066657, A066658.
%H A066720 David Applegate, <a href="http://www.research.att.com/~njas/sequences/a066721.txt">First
10000 terms of A066721 and their factorizations</a>
%E A066720 Warning! The terms after the 30000-th entry in David Applegate's file may be wrong. This was caused by a disk overflowing. He is re-running his program.
%H A066720 Robert E. Sawyer, <A href="http://groups.google.com/groups?hl=en&th=a534bab788d7ec46&rnum=1">Posting to sci.math newsgroup, Jan 12, 2002</a>
%t A066720 s={1}; xok:=Module[{},For[i=1,i<=n,i++,For[j=1;k=Length[dl=Divisors[s[[i]]x]],j<=k,j++;k--,If[MemberQ[s,dl[[j]]]&&MemberQ[s,dl[[k]]],Return[False]]]];True]; For[n=1,True,n++,Print[s[[n]]];For[x=s[[n]]+1,True,x++,If[xok,AppendTo[s,x];Break[]]]] (from Dean Hickerson)
%t A066720 a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, b = c = Table[a[i], {i, 1, n - 1}], d}, While[c = Append[b, k]; Length[ Union[ Flatten[ Table[ c[[i]]/c[[j]], {i, 1, n}, {j, 1, n}]]]] != n^2 - n + 1, k++ ]; Return[k]]; Table[ a[n], {n, 1, 75} ] (from Robert G. Wilson v)
%o A066720 (PARI) {a066720(m) = local(a,n,s,b,v,bi,bj,i,j,k);a=[1]; for(n=2,m,s=a[n-1]+1;a=concat(a,s);b=1;while (b,v=Set([]);bi=1;i=1;while(bi&&i<=n,bj=1;j=1; while(bj&&j<=n,if(i!=j,k=a[i]/a[j]; if(setsearch(v,k)==0,v=setunion(v,Set([k])),bj=0;bi=0;s++;a[n]=s;));j++);i++); if(i>n,b=0)));a}
%Y A066720 Consists of the primes together with A066721. Cf. A005282, A066775.
%Y A066720 For the rationals that are produced see A066657/A066658 and A066848, A066849.
%K A066720 nonn,nice
%O A066720 1,2
%A A066720 njas, Jan 15 2002
%E A066720 More terms from Dean Hickerson (dean at math.ucdavis.edu), Klaus Brockhaus (klaus-brockhaus at t-online.de) and David Applegate (david at research.att.com), Jan 15 2002

THE BIG QUESTION:
%C A066720 Does every rational number appear as a ratio? See A066657, A066658.
The answer is not known!

o We wrote up what we can prove about the EKG sequence, A064413,
     a favorite from a few months ago.  The paper is at the top of my home page,
     http://www.research.att.com/~njas/

Neil Sloane