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.

Matti Jutila, On numbers with a large prime factor II, J. Indian Math. Soc. (N.S.) (1974), 125--130.

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.

Thomas Bloom, Problem #961, Erdős Problems.

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

Paul Erdős, On consecutive integers, Nieuw Arch. Wisk. 3 (1955), 124-128.

Paul Erdős, On consecutive integers, Nieuw Arch. Wisk. 3 (1955), 124--128.

Paul Erdős, A Theorem of Sylvester and Schur, J. London Math. Soc. (1934), 282--288.

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

K. Ramachandra and T. Shorey, On gaps between numbers with a large prime factor, Acta Arithmetica 24.1 (1973): 99-111.

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

From Elijah Beregovsky, Jan 08 2026: (Start)

Exists c > 0 such that a(n) < c * n/log(n) (Erdős).

a(n) << log(log(log(n)))/log(log(n)) * n/log(n) (Jutila, Ramachandra & Shorey). (End)

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: A394431 A205324 A359512 * A132983 A348182 A029133

Adjacent sequences: A213250 A213251 A213252 * A213254 A213255 A213256

Keyword

nonn,changed

Author

N. J. A. Sloane, Jun 07 2012

Status

approved