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A326078
Number of subsets of {2..n} containing all of their integer quotients > 1.
8
1, 1, 2, 4, 8, 16, 24, 48, 72, 144, 216, 432, 552, 1104, 1656, 2592, 3936, 7872, 10056, 20112, 26688, 42320, 63480, 126960, 154800, 309600, 464400, 737568, 992160, 1984320, 2450880, 4901760, 6292800, 10197312, 15295968, 26241696, 32947488, 65894976, 98842464, 161587872, 205842528
OFFSET
0,3
COMMENTS
These sets are closed under taking the quotient of two distinct divisible terms.
FORMULA
For n > 0, a(n) = A326023(n) - 1.
For n > 0, a(n) = A326079(n)/2.
EXAMPLE
The a(6) = 24 subsets:
{} {2} {2,3} {2,3,4} {2,3,4,5} {2,3,4,5,6}
{3} {2,4} {2,3,5} {2,3,4,6}
{4} {2,5} {2,3,6} {2,3,5,6}
{5} {3,4} {2,4,5}
{6} {3,5} {3,4,5}
{4,5} {4,5,6}
{4,6}
{5,6}
MATHEMATICA
Table[Length[Select[Subsets[Range[2, n]], SubsetQ[#, Divide@@@Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]]&]], {n, 0, 10}]
PROG
(PARI)
a(n)={
my(lim=vector(n, k, sqrtint(k)));
my(accept(b, k)=for(i=2, lim[k], if(k%i ==0 && bittest(b, i) != bittest(b, k/i), return(0))); 1);
my(recurse(k, b)=
my(m=1);
for(j=max(2*k, n\2+1), min(2*k+1, n), if(accept(b, j), m*=2));
k++;
m*if(k > n\2, 1, (self()(k, b) + if(accept(b, k), self()(k, b + (1<<k)))))
);
recurse(1, 0);
} \\ Andrew Howroyd, Aug 30 2019
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 05 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Aug 30 2019
STATUS
approved