OFFSET
0,2
COMMENTS
a(n) counts all subsequences of {1, ..., n} in which no term divides any other. If n is a prime a(n) = 2*a(n-1)-1 because for each subsequence s counted by a(n-1) two different subsequences are counted by a(n): s and s,n. There is only one exception: 1,n is not a primitive subsequence because 1 divides n. For all n>1: a(n) < 2*a(n-1). - Alois P. Heinz, Mar 07 2011
Maximal primitive subsets are counted by A326077. - Gus Wiseman, Jun 07 2019
REFERENCES
Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 320. - N. J. A. Sloane, Apr 06 2012
LINKS
Juliana Couras, Ricardo Jesus, and Tomás Oliveira e Silva, Table of n, a(n) for n = 0..800 (terms up to n=80 from Alois P. Heinz)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Nathan McNew, Counting primitive subsets and other statistics of the divisor graph of {1,2,..,n}, arXiv:1808.04923 [math.NT], 2018.
Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
Eric Weisstein's World of Mathematics, Primitive Sequence
EXAMPLE
a(4) = 7, the primitive subsequences (including the empty sequence) are: (), (1), (2), (3), (4), (2,3), (3,4).
a(5) = 13 = 2*7-1, the primitive subsequences are: (), (5), (1), (2), (2,5), (3), (3,5), (4), (4,5), (2,3), (2,3,5), (3,4), (3,4,5).
From Gus Wiseman, Jun 07 2019: (Start)
The a(0) = 1 through a(5) = 13 primitive (pairwise indivisible) subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{3} {3} {3}
{2,3} {4} {4}
{2,3} {5}
{3,4} {2,3}
{2,5}
{3,4}
{3,5}
{4,5}
{2,3,5}
{3,4,5}
a(n) is also the number of subsets of {1..n} containing all of their pairwise products <= n as well as any quotients of divisible elements. For example, the a(0) = 1 through a(5) = 13 subsets are:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{1,2} {1,2} {1,3} {1,3}
{1,3} {1,4} {1,4}
{1,2,3} {1,2,4} {1,5}
{1,3,4} {1,2,4}
{1,2,3,4} {1,3,4}
{1,3,5}
{1,4,5}
{1,2,3,4}
{1,2,4,5}
{1,3,4,5}
{1,2,3,4,5}
Also the number of subsets of {1..n} containing all of their multiples <= n. For example, the a(0) = 1 through a(5) = 13 subsets are:
{} {} {} {} {} {}
{1} {2} {2} {3} {3}
{1,2} {3} {4} {4}
{2,3} {2,4} {5}
{1,2,3} {3,4} {2,4}
{2,3,4} {3,4}
{1,2,3,4} {3,5}
{4,5}
{2,3,4}
{2,4,5}
{3,4,5}
{2,3,4,5}
{1,2,3,4,5}
(End)
From Gus Wiseman, Mar 12 2024: (Start)
Also the number of subsets of {1..n} containing all divisors of the elements. For example, the a(0) = 1 through a(6) = 17 subsets are:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{1,2} {1,2} {1,2} {1,2}
{1,3} {1,3} {1,3}
{1,2,3} {1,2,3} {1,5}
{1,2,4} {1,2,3}
{1,2,3,4} {1,2,4}
{1,2,5}
{1,3,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,2,3,4,5}
(End)
MAPLE
with(numtheory):
b:= proc(s) option remember; local n;
n:= max(s[]);
`if`(n<0, 1, b(s minus {n}) + b(s minus divisors(n)))
end:
bb:= n-> b({$2..n} minus divisors(n)):
sb:= proc(n) option remember; `if`(n<2, 0, bb(n) + sb(n-1)) end:
a:= n-> `if`(n=0, 1, `if`(isprime(n), 2*a(n-1)-1, 2+sb(n))):
seq(a(n), n=0..40); # Alois P. Heinz, Mar 07 2011
MATHEMATICA
b[s_] := b[s] = With[{n=Max[s]}, If[n < 0, 1, b[Complement[s, {n}]] + b[Complement[s, Divisors[n]]]]];
bb[n_] := b[Complement[Range[2, n], Divisors[n]]];
sb[n_] := sb[n] = If[n < 2, 0, bb[n] + sb[n-1]];
a[n_] := If[n == 0, 1, If[PrimeQ[n], 2a[n-1] - 1, 2 + sb[n]]]; Table[a[n], {n, 0, 37}]
(* Jean-François Alcover, Jul 27 2011, converted from Maple *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Select[Union@@Table[#*i, {i, n}], #<=n&]]&]], {n, 10}] (* Gus Wiseman, Jun 07 2019 *)
Table[Length[Select[Subsets[Range[n]], #==Union@@Divisors/@#&]], {n, 0, 10}] (* Gus Wiseman, Mar 12 2024 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from David Wasserman, May 02 2002
a(32)-a(37) from Donovan Johnson, Aug 11 2010
STATUS
approved