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A058989 Largest number of consecutive integers such that each is divisible by a prime <= the n-th 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, 875, 907, 925, 953, 977, 1001, 1029, 1057, 1097, 1109 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Marty Weissman conjectured that a(n)=2q-1, where q is the largest prime smaller than the n-th prime. The conjecture holds for the first few terms, but then a(n) is larger than 2q-1. Phil Carmody proved a(n)>=2q-1. 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
Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
H. Iwaniec, On the error term in the linear sieve, Acta. Arith. 19 (1971), pp. 1-30.
J. D. Laison and M. Schick, Seeing dots: visibility of lattice points, Mathematics Magazine, Vol. 80 (2007), pp. 274-282. See page 281 reference 13.
János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.
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 = k-L]; 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}] (* Jean-François Alcover, Sep 05 2017 *)
CROSSREFS
Sequence in context: A178415 A249424 A076274 * A049691 A206297 A320596
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
a(55)-a(64) from A048670 added by Constantino Calancha, Aug 05 2023
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)