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A059686
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Grimm numbers (1): a(n) = largest k so that for each m in {n+1, n+2, ..., n+k} there corresponds a different prime factor p_m.
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4
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2, 3, 4, 4, 3, 5, 4, 6, 6, 7, 6, 7, 6, 5, 8, 8, 7, 8, 7, 7, 8, 7, 6, 7, 9, 8, 8, 11, 10, 11, 10, 11, 11, 10, 12, 12, 11, 10, 9, 9, 8, 11, 10, 9, 10, 9, 8, 11, 13, 13, 12, 11, 10, 11, 14, 15, 14, 13, 12, 14, 13, 12, 13, 13, 14, 14, 13, 12, 11, 10, 9, 15, 14, 13, 14, 13, 13, 17, 16, 17
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| R. K. Guy, Unsolved Problems in Theory of Numbers, 2nd ed., Section B32, discusses some conjectures of Grimm that could produce related sequences.
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REFERENCES
| C. A. Grimm, A conjecture on consecutive composite numbers, Amer. Math. Monthly, 76 (1969), 1126-1128.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XII.15, p. 438.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
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EXAMPLE
| For n=4 we look at the sequence {5,6,7,8,9,...} and we must pick a different prime factor for as many as we can. We can choose 5 for 5, 3 for 6, 7 for 7, 2 for 8, but now we are stuck, so k=4, a(4) = 4.
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MATHEMATICA
| Needs["DiscreteMath`Combinatorica`"]; factors[n_Integer] := First[Transpose[FactorInteger[n]]]; Join[{2, 3}, Table[k=2; While[s=Table[{}, {n0+k}]; prms=0; Do[If[PrimeQ[n], prms++, t=factors[n]; s[[n]]=t; Do[i=t[[j]]; If[i<n, AppendTo[s[[i]], n]], {j, Length[t]}]], {n, n0+1, n0+k}]; Length[BipartiteMatching[FromAdjacencyLists[s]]]+prms == k, k++ ]; k-1, {n0, 3, 80}]] (T. D. Noe)
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CROSSREFS
| Cf. A059751, A059752.
Cf. A101083 (largest k such that the product (n+1)(n+2)...(n+k) has at least k distinct prime factors).
Sequence in context: A129456 A030412 A160371 * A101083 A097935 A188940
Adjacent sequences: A059683 A059684 A059685 * A059687 A059688 A059689
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Feb 06 2001
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EXTENSIONS
| More terms from Fabian Rothelius (fabian.rothelius(AT)telia.com), Feb 08 2001. Corrected and extended by Naohiro Nomoto (6284968128(AT)geocities.co.jp), Feb 28 2001.
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