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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. 8

%I

%S 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 73,79,81,83,89,97,98,101,103,105,107,109,113,127,128,131,137,139,149,

%U 151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239

%N 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 Sequence was apparently invented by Jeromino Wannhoff - see the Rosenthal link.

%C 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 Does every rational number appear as a ratio? See A066657, A066658.

%H David Applegate, <a href="/A066721/a066721.txt">First 48186 terms of A066721 and their factorizations</a> (implies first 8165063 terms of current sequence)

%H Rainer Rosenthal, <a href="https://groups.google.com/forum/#!searchin/de.rec.denksport/%22%22/de.rec.denksport/6AHOtVMj1iI/nPNhu3mmWjEJ">Posting to de.rec.denksport, Jan 15 2002</a>

%H Robert E. Sawyer, <a href="https://groups.google.com/forum/?hl=en#!searchin/sci.math/%22%22/sci.math/0SPZNahXf-Y/CCHVZrIeswIJ">Is there such a sequence?</a> Posting by r.e.s. to sci.math newsgroup, Jan 13, 2002

%t 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[]]]] (* _Dean Hickerson_ *)

%t 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} ] (* _Robert G. Wilson v_ *)

%o (PARI) {a066720(m) = local(a,rat,n,s,new,b,i,k,j); a=[]; rat=Set([]); n=0; s=0; while(s<m,s++; new=Set([]); b=1; i=1; while(b&&i<=n,k=s/a[i]; if(setsearch(rat,k),b=0,new=setunion(new,Set(k)); k=a[i]/s; if(setsearch(rat,k),b=0,new=setunion(new,Set(k)))); i++); if(b,rat=setunion(rat,new); a=concat(a,s); n++; print1(s,",")))} a066720(240) \\ _Klaus Brockhaus_, Feb 23 2002

%o (Haskell)

%o import qualified Data.Set as Set (null)

%o import Data.Set as Set (empty, insert, member)

%o a066720 n = a066720_list !! (n-1)

%o a066720_list = f [] 1 empty where

%o f ps z s | Set.null s' = f ps (z + 1) s

%o | otherwise = z : f (z:ps) (z + 1) s'

%o where s' = g (z:ps) s

%o g [] s = s

%o g (x:qs) s | (z * x) `member` s = empty

%o | otherwise = g qs $ insert (z * x) s

%o -- _Reinhard Zumkeller_, Nov 19 2013

%Y Consists of the primes together with A066721. Cf. A005282, A066775.

%Y For the rationals that are produced see A066657/A066658 and A066848, A066849.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_, Jan 15 2002

%E More terms from _Dean Hickerson_, _Klaus Brockhaus_ and _David Applegate_, Jan 15 2002

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Last modified May 24 20:53 EDT 2019. Contains 323534 sequences. (Running on oeis4.)