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Search: id:A066720
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| A066720 |
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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. |
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7
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| 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, 73, 79, 81, 83, 89, 97, 98, 101, 103, 105, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Sequence was apparently invented by Jeromino Wannhoff - see the Rosenthal link.
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
Does every rational number appear as a ratio? See A066657, A066658.
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LINKS
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David Applegate, First 48186 terms of A066721 and their factorizations (implies first 8165063 terms of current sequence)
Rainer Rosenthal, Posting to de.rec.denksport, Jan 15 2002
Robert E. Sawyer, Posting to sci.math newsgroup, Jan 12, 2002
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MATHEMATICA
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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)
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)
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PROGRAM
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(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) (from Klaus Brockhaus, Feb 23 2002)
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CROSSREFS
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Consists of the primes together with A066721. Cf. A005282, A066775.
For the rationals that are produced see A066657/A066658 and A066848, A066849.
Adjacent sequences: A066717 A066718 A066719 this_sequence A066721 A066722 A066723
Sequence in context: A014121 A051600 A026410 this_sequence A060634 A026401 A069353
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KEYWORD
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nonn,nice
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AUTHOR
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njas, Jan 15 2002
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EXTENSIONS
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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
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