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A213253 a(n) = smallest k such that highest prime factor of m(m+1)...(m+k-1) is > n if m > n. 2

%I #46 Jun 04 2019 17:50:58

%S 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,

%T 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,

%U 14,14,14,14,14,14,14,14,14,14,14

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

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

%C For Erdos and Ecklund-Eggleton's stronger theorem, see A220314 - _Jonathan Sondow_, Dec 10 2012

%C 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

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

%D 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.

%H T. D. Noe, <a href="/A213253/b213253.txt">Table of n, a(n) for n = 1..268</a> (from Najman's paper)

%H M. Bauer and M. A. Bennett, <a href="https://doi.org/10.1090/S0025-5718-08-02134-0">Prime factors of consecutive integers</a>, Math. Comp., 77 (2008), 2455-2459.

%H E. F. Ecklund and R. B. Eggleton, <a href="https://www.jstor.org/stable/2317422">Prime factors of consecutive integers</a>, Amer. Math. Monthly, 79 (1972), 1082-1089.

%H P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1955-04.pdf">On consecutive integers</a>, Nieuw Arch. Wisk. 3 (1955), 124-128.

%H Filip Najman, <a href="http://arxiv.org/abs/1108.3710">Large strings of consecutive smooth integers</a>, Arch. Math. (Basel) 97 (2011), 319-324; arXiv:1108.3710 [math.NT].

%H J. J. Sylvester, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN599484047_0021&amp;DMDID=DMDLOG_0003&amp;IDDOC=644640">On arithmetical series</a>, Messenger Math. 21 (1892), 1-19, 87-120, 192.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bertrand%27s_postulate#Sylvester.27s_theorem">Sylvester's Theorem</a>

%F a(n) <= n (Sylvester's theorem--see Sylvester 1892, p. 4) - _Jonathan Sondow_, Jul 23 2013

%t (* 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 *)

%Y Cf. A220314.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jun 07 2012

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