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A323457 Largest cardinality of any set that is "special above n".

%I A323457 #22 Jan 30 2019 10:27:53
%S A323457 1,1,2,5,9
%N A323457 Largest cardinality of any set that is "special above n".
%C A323457 A set A of positive integers is called "special above n" iff every element x > n of A divides the product of all elements y < x of A and does not divide any element y > x; an empty product is taken to be 1.
%C A323457 This is a corrected version of A191550, which was based on Friedman (2000), and has terms 1,1,2,5,8,37,26984.
%C A323457 The entries for a(4), a(5), a(6) appear to be wrong. I added the explicit example that shows a(4) >= 9 (and the proof that a(4) <= 9 is easy). I also added the estimate a(5) > 2^2^2^33. An explicit listing proving this is in the Links; that construction is due to Jim Henle. The 2^2^2^33 lower bound for a(5) makes the comment (retained) that a(7) >= 2^2^2^60 seem suspect: it is surely very much larger than this.
%C A323457 a(5) > 2^2^2^33, a(7) > 2^2^2^60, a(11) > A_3(1000), a(13) > A_4(5000), where A_n is the Ackermann function as defined by Harvey Friedman: A_1(n) = 2n, A_2(n) = 2^n, A_{k+1}(n) = A_k A_k ... A_k(1), where there are n A_k's (see also A014221).
%H A323457 Harvey M. Friedman, <a href="https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/EnormousInt.12pt.6_1_00-23kmig3.pdf">Enormous Integers in Real Life</a>, June 1, 2000, 11 pages, see p. 6.
%H A323457 Stan Wagon, <a href="/A323457/a323457_1.pdf">Proof that a(5) > 2^2^2^33</a>
%e A323457 a(2) = #{1, 2} = 2,
%e A323457 a(3) = #{1, 2, 3, 6, 9} = 5,
%e A323457 a(4) = #{1, 2, 3, 4, 24, 32, 36, 54, 81} = 9.
%e A323457 Examples to illustrate the definition of "special above n":
%e A323457 {1,2,3,4} is special above 4 but not special above 3,
%e A323457 {1,2,4,8} is special above 4 but not special above 3,
%e A323457 {1,2,3,6,12} is special above 6 but not special above 5.
%Y A323457 Cf. A191550.
%K A323457 nonn,hard
%O A323457 0,3
%A A323457 _Stan Wagon_, Jan 16 2019
%E A323457 Edited by _N. J. A. Sloane_, Jan 19 2019