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%I #30 Feb 25 2016 03:06:16
%S 1,2,12,120,5040,131040
%N Smallest tau-value of a special set (in the sense of Heath-Brown) with n elements.
%C A special set is a subset of the positive integers such that the absolute difference of any two distinct terms is the same as their greatest common divisor. Tau of a set of positive integers is defined as the least common multiple of the absolute differences in the set.
%C a(n) exists for each n, a consequence of Heath-Brown's 1984 result that there are special sets of arbitrarily large cardinality.
%D D. R. Heath-Brown, Consecutive almost-primes, J. Indian Math. Soc. (N.S.) 52 (1987), pp. 39-49 (1988).
%D D. R. Heath-Brown, A note on the paper: "Consecutive almost-primes", J. Indian Math. Soc. (N.S.) 66 no. 1-4 (1999), pp. 203-205.
%D Adolf J. Hildebrand, Multiplicative properties of consecutive integers; pp. 103-118 in Analytic number theory, ed. by Y. Motohashi.
%H D. R. Heath-Brown, <a href="http://dx.doi.org/10.1112/S0025579300010743">The divisor function at consecutive integers</a>, Mathematika 31 (1984), pp. 141-149.
%H Adolf Hildebrand, <a href="http://dx.doi.org/10.1090/S0002-9939-1985-0810155-4">On a conjecture of Balog</a>, Proceedings of the American Mathematical Society 95:4 (1985), pp. 517-523.
%F Heath-Brown (1988) proved that n log n << log a(n) << n^3 log n.
%e {1,2} is special since 2-1 = gcd(2, 1). tau({1,2}) = lcm({2-1}) = 1, so a(2) = 1.
%e {2,3,4} is special since 3-2 = gcd(3,2), 4-3 = gcd(4,3), and 4-2 = gcd(4,2). tau({2,3,4}) = lcm({3-2,4-3,4-2}) = 2, so a(3) = 2.
%e tau({1,2}) = 1.
%e tau({2,3,4}) = 2.
%e tau({8,9,10,12}) = 12.
%e tau({40,45,48,50,60}) = 120.
%e tau({210, 216, 220, 224, 225, 240}) = 5040.
%e tau({49920, 49950, 49952, 49959, 49960, 49968, 49980}) = 131040.
%K nonn,hard,more
%O 2,2
%A _Charles R Greathouse IV_, Jan 10 2013
%E a(6) from _Charles R Greathouse IV_, Jan 20 2013
%E a(7) from _Charles R Greathouse IV_, Feb 07 2013
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