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%I #143 Aug 25 2021 01:07:52
%S 2,4,6,10,14,22,26,34,40,46,58,66,74,90,100,106,118,132,152,174,190,
%T 200,216,234,258,264,282,300,312,330,354,378,388,414,432,450,476,492,
%U 510,538,550,574,600,616,642,660,686,718,742,762,798,810,834,858,876,908,926,954
%N Jacobsthal function A048669 applied to the product of the first n primes (A002110).
%C 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
%C 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
%C 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
%C Computation of a(62)-a(64) was supported by Google Cloud. - _Andrzej Bozek_, Mar 14 2021
%D L. E. Dickson, History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.
%H Andrzej Bozek, <a href="/A048670/b048670.txt">Table of n, a(n) for n = 1..64</a>
%H Andrzej Bozek, <a href="/A048670/a048670_5.txt">Gerbicz's table with added a(58)-a(64)</a>
%H Fintan Costello and Paul Watts, <a href="https://arxiv.org/abs/1208.5342">A computational upper bound on Jacobsthal's function</a>, arXiv:1208.5342 [math.NT], 2012.
%H Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, <a href="https://arxiv.org/abs/1412.5029">Long gaps between primes</a>, arXiv:1412.5029 [math.NT], 2014-2016; Journal of the American Mathematical Society 31:1 (2018), pp. 65-105.
%H Robert Gerbicz, <a href="/A048670/a048670.txt">Table of n, a(n), u(n) for n=1..57</a>, 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.
%H Thomas R. Hagedorn, <a href="https://doi.org/10.1090/S0025-5718-08-02166-2">Computation of Jacobsthal's function h(n) for n < 50</a>, Math. Comp. 78 (2009) 1073-1087.
%H L. Hajdu and N. Saradha, <a href="https://doi.org/10.1090/S0025-5718-2012-02581-6">Disproof of a conjecture of Jacobsthal</a>, Mathematics of Computation 81 (2012), pp. 2461-2471.
%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 Arithmetica 19 (1971), pp. 1-30.
%H Helmut Maier and Carl Pomerance, <a href="https://doi.org/10.1090/S0002-9947-1990-0972703-X">Unusually large gaps between consecutive primes</a>, Transactions of the American Mathematical Society 322:1 (1990), pp. 201-237.
%H János Pintz, <a href="https://doi.org/10.1006/jnth.1997.2081">Very large gaps between consecutive primes</a>, Journal of Number Theory 63 (1997), pp. 286-301.
%H Mario Ziller, <a href="https://arxiv.org/abs/1903.11973">New computational results on a conjecture of Jacobsthal</a>, arXiv:1903.11973 [math.NT], 2019.
%H Mario Ziller, <a href="https://arxiv.org/abs/2007.01808">On differences between consecutive numbers coprime to primorials</a>, arXiv:2007.01808 [math.NT], 2020.
%H Mario Ziller and 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) = A058989(n) + 1.
%F a(n) << n^2*(log n)^2, see Iwaniec. - _Charles R Greathouse IV_, Sep 08 2012
%F a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see Pintz.
%F a(n) = 2 * A072752(n) + 2. - _Mario Ziller_, Dec 08 2016
%F 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
%F a(n) = largest (or last) term in row n of A331118. - _Michael De Vlieger_, Dec 11 2020
%t (* 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_ *)
%Y Cf. A048669, A002110, A005867, A058989, A072752, A331118.
%K nonn,nice,hard
%O 1,1
%A _Jan Kristian Haugland_
%E a(21)-a(24) from _Max Alekseyev_, Apr 09 2006
%E a(25)-a(49) from _Thomas Hagedorn_, Feb 21 2007
%E 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
%E a(55)-a(57) from _Robert Gerbicz_, Apr 10 2017
%E a(58)-a(64) from _Andrzej Bozek_, Aug 2020 - Mar 2021
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