

A058989


Largest number of consecutive integers such that each is divisible by a prime <= the nth prime.


6



1, 3, 5, 9, 13, 21, 25, 33, 39, 45, 57, 65, 73, 89, 99, 105, 117, 131, 151, 173, 189, 199, 215, 233, 257, 263, 281, 299, 311, 329, 353, 377, 387, 413, 431, 449, 475, 491, 509, 537, 549, 573, 599, 615, 641, 659, 685, 717, 741, 761, 797, 809, 833, 857
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OFFSET

1,2


COMMENTS

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.
A049300(n) is the smallest value of the mentioned consecutive integers.  Reinhard Zumkeller, Jun 14 2003


REFERENCES

Dickson, L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.


LINKS

Table of n, a(n) for n=1..54.
Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 10731087.
H. Iwaniec, On the error term in the linear sieve, Acta. Arith. 19 (1971), pp. 130.
J. D. Laison and M. Schick, Seeing dots: visibility of lattice points, Mathematics Magazine, Vol. 80 (2007), pp. 274282. See page 281 reference 13.
János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286301.
Mario Ziller, John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.


FORMULA

a(n) = A048670(n)  1. See that entry for additional information.
Iwaniec proved that a(n) << n^2*(log n)^2.  Charles R Greathouse IV, Sep 08 2012
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
a(n) = 2 * A072752(n) + 1.  Mario Ziller, Dec 08 2016
See A048669 for many other bounds and references.  N. J. A. Sloane, Apr 19 2017


EXAMPLE

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.


MATHEMATICA

(* This program is not suitable to compute more than a few terms *)
primorial[n_] := Product[Prime[k], {k, 1, n}];
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];
a[1] = 1;
a[n_] := a[n] = j[primorial[n]]  1;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* JeanFrançois Alcover, Sep 05 2017 *)


CROSSREFS

Cf. A048669, A048670, A072752.
Sequence in context: A178415 A249424 A076274 * A049691 A206297 A320596
Adjacent sequences: A058986 A058987 A058988 * A058990 A058991 A058992


KEYWORD

nice,nonn


AUTHOR

Jud McCranie, Jan 16 2001


EXTENSIONS

Laison and Schick reference from Parthasarathy Nambi, Oct 19 2007
More terms from A048670 added by Max Alekseyev, Feb 07 2008
a(46) corrected and a(50)a(54) added by Mario Ziller, Dec 08 2016


STATUS

approved



