%I
%S 1,3,5,9,13,21,25,33,39,45,57,65,73,89,99,105,117,131,151,173,189,199,
%T 215,233,257,263,281,299,311,329,353,377,387,413,431,449,475,491,509,
%U 537,549,573,599,615,641,659,685,717,741,761,797,809,833,857
%N Largest number of consecutive integers such that each is divisible by a prime <= the nth prime.
%C Marty Weissman conjectured that a(n)=2q1, where q is the largest prime smaller than the nth prime. The conjecture holds for the first few terms, but then a(n) is larger than 2q1. Phil Carmody proved a(n)>=2q1. Terms were calculated by Weissman, Carmody and McCranie.
%C A049300(n) is the smallest value of the mentioned consecutive integers.  _Reinhard Zumkeller_, Jun 14 2003
%D Dickson, L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.
%H Thomas R. Hagedorn, <a href="https://doi.org/10.1090/S0025571808021662">Computation of Jacobsthal's function h(n) for n < 50</a>, Math. Comp. 78 (2009) 10731087.
%H H. Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa19/aa1911.pdf">On the error term in the linear sieve</a>, Acta. Arith. 19 (1971), pp. 130.
%H J. D. Laison and M. Schick, <a href="http://www.jstor.org/stable/27643042">Seeing dots: visibility of lattice points</a>, Mathematics Magazine, Vol. 80 (2007), pp. 274282. See page 281 reference 13.
%H János Pintz, <a href="http://dx.doi.org/10.1006/jnth.1997.2081">Very large gaps between consecutive primes</a>, Journal of Number Theory 63 (1997), pp. 286301.
%H Mario Ziller, John F. Morack, <a href="https://arxiv.org/abs/1611.03310">Algorithmic concepts for the computation of Jacobsthal's function</a>, arXiv:1611.03310 [math.NT], 2016.
%F a(n) = A048670(n)  1. See that entry for additional information.
%F Iwaniec proved that a(n) << n^2*(log n)^2.  _Charles R Greathouse IV_, Sep 08 2012
%F a(n) >= (2e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see A048670.  _Charles R Greathouse IV_, Sep 08 2012
%F a(n) = 2 * A072752(n) + 1.  _Mario Ziller_, Dec 08 2016
%F See A048669 for many other bounds and references.  _N. J. A. Sloane_, Apr 19 2017
%e The 4th prime is 7. Nine is the maximum number of consecutive integers such that each is divisible by 2, 3, 5 or 7. (Example: 2 through 10) So a(4)=9.
%t (* This program is not suitable to compute more than a few terms *)
%t primorial[n_] := Product[Prime[k], {k, 1, n}];
%t j[n_] := Module[{L = 1, m = 1}, For[k = 2, k <= n+1, k++, If[GCD[k, n] == 1, If[L+m < k, m = kL]; L = k]]; m];
%t a[1] = 1;
%t a[n_] := a[n] = j[primorial[n]]  1;
%t Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* _JeanFrançois Alcover_, Sep 05 2017 *)
%Y Cf. A048669, A048670, A072752.
%K nice,nonn
%O 1,2
%A _Jud McCranie_, Jan 16 2001
%E Laison and Schick reference from _Parthasarathy Nambi_, Oct 19 2007
%E More terms from A048670 added by _Max Alekseyev_, Feb 07 2008
%E a(46) corrected and a(50)a(54) added by _Mario Ziller_, Dec 08 2016
