login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A213253 a(n) = smallest k such that highest prime factor of m(m+1)...(m+k-1) 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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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.

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

LINKS

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

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-Fran├žois Alcover, Nov 25 2013 *)

CROSSREFS

Cf. A220314.

Sequence in context: A063826 A152983 A205324 * A132983 A029133 A230411

Adjacent sequences:  A213250 A213251 A213252 * A213254 A213255 A213256

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 07 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 19 00:07 EDT 2014. Contains 240735 sequences.