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

1, 1, 2, 5, 9

Offset: 0

Author

Stan Wagon, Jan 16 2019

Keywords

• nonn
• hard

Comments

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.

This is a corrected version of A191550, which was based on Friedman (2000), and has terms 1,1,2,5,8,37,26984.

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.

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).

Examples

			a(2) = #{1, 2} = 2,
a(3) = #{1, 2, 3, 6, 9} = 5,
a(4) = #{1, 2, 3, 4, 24, 32, 36, 54, 81} = 9.
Examples to illustrate the definition of "special above n":
{1,2,3,4} is special above 4 but not special above 3,
{1,2,4,8} is special above 4 but not special above 3,
{1,2,3,6,12} is special above 6 but not special above 5.
		

Links

• Harvey M. Friedman, Enormous Integers in Real Life, June 1, 2000, 11 pages, see p. 6.
• Stan Wagon, Proof that a(5) > 2^2^2^33

Crossrefs

Cf. A191550.

Extensions

Edited by N. J. A. Sloane, Jan 19 2019