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A048670
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Jacobsthal function A048669 applied to the product of the first n primes (A002110).
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14
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2, 4, 6, 10, 14, 22, 26, 34, 40, 46, 58, 66, 74, 90, 100, 106, 118, 132, 152, 174, 190, 200, 216, 234, 258, 264, 282, 300, 312, 330, 354, 378, 388, 414, 432, 450, 476, 492, 510, 538, 550, 574, 600, 616, 642, 660, 686, 718, 742, 762, 798, 810, 834, 858, 876, 908, 926, 954
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.
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LINKS
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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.
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FORMULA
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a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see Pintz.
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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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nonn,nice,hard
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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