OFFSET
1,1
COMMENTS
Pintz shows that j(x#) >= (2*e^gamma + o(1)) x log x log log log x / (log log x)^2 and hence a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2 by the Prime Number Theorem. - Charles R Greathouse IV, Sep 08 2012
Jacobsthal conjectures that a(n) >= j(k) := A048669(k) for any k with n prime factors, which would make this the RECORDS transform of A048669. Hajdu & Saradha disprove the conjecture, showing that this fails for n = 24 where j(k) = 236 > 234 = a(24) for any k divisible by 76964283982898776138308824190 and with 24 prime factors in total. - Charles R Greathouse IV, Sep 08 2012 / Edited by Jan Kristian Haugland, Feb 02 2019
Ford, Green, Konyagin, Maynard, & Tao show that j(x#) >> x log x log log log x / log log x and hence a(n) >> n log^2 n log log log n / log log n. - Charles R Greathouse IV, Mar 29 2018
Computation of a(62)-a(64) was supported by Google Cloud. - Andrzej Bozek, Mar 14 2021
REFERENCES
L. E. Dickson, History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.
LINKS
Andrzej Bozek, Table of n, a(n) for n = 1..64
Andrzej Bozek, Gerbicz's table with added a(58)-a(64)
Fintan Costello and Paul Watts, A computational upper bound on Jacobsthal's function, arXiv:1208.5342 [math.NT], 2012.
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Long gaps between primes, arXiv:1412.5029 [math.NT], 2014-2016; Journal of the American Mathematical Society 31:1 (2018), pp. 65-105.
Robert Gerbicz, Table of n, a(n), u(n) for n=1..57, where every integer from [u(n),u(n)+a(n)-2] is divisible by at least one of the first n primes. Note that u(n) is not unique, the smallest is given by A049300(n).
Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
L. Hajdu and N. Saradha, Disproof of a conjecture of Jacobsthal, Mathematics of Computation 81 (2012), pp. 2461-2471.
H. Iwaniec, On the error term in the linear sieve, Acta Arithmetica 19 (1971), pp. 1-30.
Helmut Maier and Carl Pomerance, Unusually large gaps between consecutive primes, Transactions of the American Mathematical Society 322:1 (1990), pp. 201-237.
János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.
Mario Ziller, New computational results on a conjecture of Jacobsthal, arXiv:1903.11973 [math.NT], 2019.
Mario Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT], 2020.
Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.
FORMULA
a(n) = A058989(n) + 1.
a(n) << n^2*(log n)^2, see Iwaniec. - Charles R Greathouse IV, Sep 08 2012
a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see Pintz.
a(n) = 2 * A072752(n) + 2. - Mario Ziller, Dec 08 2016
Maier & Pomerance conjecture that Max_{n <= x} A048669(n) = log(x)*(log log x)^(2+o(1)) which suggests a(n) = n*(log n)^(3+o(1)). - Charles R Greathouse IV, Mar 29 2018
a(n) = largest (or last) term in row n of A331118. - Michael De Vlieger, Dec 11 2020
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[n_] := a[n] = j[primorial[n]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 27 2013, after M. F. Hasler *)
CROSSREFS
KEYWORD
nonn,nice,hard
AUTHOR
EXTENSIONS
a(21)-a(24) from Max Alekseyev, Apr 09 2006
a(25)-a(49) from Thomas Hagedorn, Feb 21 2007
a(46) corrected (published value in Hagedorn's 2009 Mathematics of Computation article was correct) and a(50)-a(54) added by Mario Ziller, Dec 08 2016
a(55)-a(57) from Robert Gerbicz, Apr 10 2017
a(58)-a(64) from Andrzej Bozek, Aug 2020 - Mar 2021
STATUS
approved