

A213253


a(n) = smallest k such that highest prime factor of m(m+1)...(m+k1) is > n if m > n.


2



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
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OFFSET

1,2


COMMENTS

By a theorem of Sylvester, a(n) always exists.
For Erdos and EcklundEggleton'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), 155157.
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), 161162.


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), 24552459.
E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 10821089.
P. Erdos, On consecutive integers, Nieuw Arch. Wisk. 3 (1955), 124128.
Filip Najman, Large strings of consecutive smooth integers, Arch. Math. (Basel) 97 (2011), 319324; arXiv:1108.3710 [math.NT].
J. J. Sylvester, On arithmetical series, Messenger Math. 21 (1892), 119, 87120, 192.
Wikipedia, Sylvester's Theorem


FORMULA

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


MATHEMATICA

(* To speed up computation, it is assumed that a(n) >= a(n1)2 and m <= n^2 *) a[1] = 1; a[n_] := a[n] = For[k = a[n1]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}] (* JeanFrançois Alcover, Nov 25 2013 *)


CROSSREFS

Cf. A220314.
Sequence in context: A331377 A327551 A205324 * A132983 A029133 A255402
Adjacent sequences: A213250 A213251 A213252 * A213254 A213255 A213256


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jun 07 2012


STATUS

approved



