A096826 - OEIS

%I #12 Mar 06 2020 15:48:25

%S 1,2,2,3,2,1,2,4,3,1,2,3,2,1,1,5,2,3,2,3,1,1,2,6,3,1,4,3,2,2,2,6,1,1,

%T 1,1,2,1,1,6,2,2,2,3,3,1,2,10,3,3,1,3,2,6,1,6,1,1,2,1,2,1,3,7,1,2,2,3,

%U 1,2,2,4,2,1,3,3,1,2,2,10,5,1,2,1,1,1

%N Number of maximal-sized antichains in divisor lattice D(n).

%C The divisor lattice D(n) is the lattice of the divisors of the natural number n.

%H Eric M. Schmidt, <a href="/A096826/b096826.txt">Table of n, a(n) for n = 1..10000</a>

%e From _Gus Wiseman_, Aug 24 2018: (Start)

%e The a(120) = 6 antichains:

%e   {8,12,20,30}

%e   {8,12,15,20}

%e   {8,10,12,15}

%e   {6,8,15,20}

%e   {6,8,10,15}

%e   {4,6,10,15}

%e (End)

%o (Sage)

%o def A096826(n) :

%o     if n==1 : return 1

%o     R.<t> = QQ[]; mults = [x[1] for x in factor(n)]

%o     maxsize = prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]

%o     dlat = LatticePoset((divisors(n), attrcall("divides")))

%o     count = 0

%o     for ac in dlat.antichains_iterator() :

%o         if len(ac) == maxsize : count += 1

%o     return count

%o # _Eric M. Schmidt_, May 13 2013

%Y Cf. A000005, A008480, A096825, A096827, A253249, A285572 A285573.

%K nonn

%O 1,2

%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2004

%E More terms from _Eric M. Schmidt_, May 13 2013