A213253 a(n) = smallest k such that highest prime factor of m(m+1)...(m+k-1) is > n if m > n.

1, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset

1,2

Comments

By a theorem of Sylvester, a(n) always exists.

For Erdos and Ecklund-Eggleton's stronger theorem, see A220314 - Jonathan Sondow, Dec 10 2012

Najman says that standard heuristics for the size of gaps between consecutive primes lead one to expect that the order of magnitude of a(n) is (log n)^2. - Jonathan Sondow, Jul 23 2013

References

E. F. Ecklund, Jr., R. B. Eggleton and J. L. Selfridge, Factors of consecutive integers, Proc. Man. Conference Numerical Maths., Winnipeg, (1971), 155-157.

E. F. Ecklund, Jr., R. B. Eggleton and J. L. Selfridge, Consecutive integers all of whose prime factors belong to a given set, Proc. Man. Conference Numerical Maths., Winnipeg (1971), 161-162.

Links

T. D. Noe, Table of n, a(n) for n = 1..268 (from Najman's paper)

M. Bauer and M. A. Bennett, Prime factors of consecutive integers, Math. Comp., 77 (2008), 2455-2459.

E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 1082-1089.

P. Erdos, On consecutive integers, Nieuw Arch. Wisk. 3 (1955), 124-128.

Filip Najman, Large strings of consecutive smooth integers, Arch. Math. (Basel) 97 (2011), 319-324; arXiv:1108.3710 [math.NT].

J. J. Sylvester, On arithmetical series, Messenger Math. 21 (1892), 1-19, 87-120, 192.

Wikipedia, Sylvester's Theorem

Formula

a(n) <= n (Sylvester's theorem--see Sylvester 1892, p. 4) - Jonathan Sondow, Jul 23 2013

Mathematica

(* To speed up computation, it is assumed that a(n) >= a(n-1)-2 and m <= n^2 *) a[1] = 1; a[n_] := a[n] = For[k = a[n-1]-2, True, k++, If[And @@ (FactorInteger[ Pochhammer[#, k]][[-1, 1]] > n & /@ Range[n+1, n^2]), Return[k]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 268}] (* Jean-François Alcover, Nov 25 2013 *)

Crossrefs

Cf. A220314.

Sequence in context: A327551 A205324 A359512 * A132983 A348182 A029133

Adjacent sequences: A213250 A213251 A213252 * A213254 A213255 A213256

Keyword

nonn

Author

N. J. A. Sloane, Jun 07 2012

Status

approved