A059686 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.

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

Offset

1,1

Comments

Guy (2004) discusses some conjectures of Grimm that could produce related sequences.

The name "Grimm numbers" refers to the American mathematician Carl Albert Grimm (1926-2018). - Amiram Eldar, Apr 23 2024

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B32, pp. 133-134.

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XII, p. 438, Section XII.15.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

C. A. Grimm, A conjecture on consecutive composite numbers, Amer. Math. Monthly, 76 (1969), 1126-1128.

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.

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 *)

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: A030412 A329526 A160371 * A101083 A373557 A097935

Adjacent sequences: A059683 A059684 A059685 * A059687 A059688 A059689

Keyword

nonn,easy,nice,look

Author

N. J. A. Sloane, Feb 06 2001

Extensions

More terms from Fabian Rothelius, Feb 08 2001

Corrected and extended by Naohiro Nomoto, Feb 28 2001

Status

approved