OFFSET
0,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
EXAMPLE
The a(0) = 1 through a(6) = 19 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{1,3} {4} {4} {4}
{1,3} {5} {5}
{2,4} {1,3} {6}
{3,4} {1,5} {1,3}
{2,4} {1,5}
{2,5} {2,4}
{3,4} {2,5}
{4,5} {3,4}
{2,4,5} {3,6}
{4,5}
{4,6}
{5,6}
{2,4,5}
{3,4,6}
{4,5,6}
An example for n = 20 is {5,6,7,9,10,12,14,15,16,19,20}, with prime indices:
5: {3}
6: {1,2}
7: {4}
9: {2,2}
10: {1,3}
12: {1,1,2}
14: {1,4}
15: {2,3}
16: {1,1,1,1}
19: {8}
20: {1,1,3}
None of these prime indices {1,2,3,4,8} belong to the subset, as required.
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], Intersection[#, PrimePi/@First/@Join@@FactorInteger/@#]=={}&]], {n, 0, 10}]
PROG
(PARI)
pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k, b)->if(k>#p, 1, my(t=self()(k+1, b)); if(!bitand(p[k], b), t+=if(bittest(d, k), self()(k+1, b+(1<<k)), t)); t))(1, 0)} \\ Andrew Howroyd, Aug 16 2019
CROSSREFS
The maximal case is A324743. The strict integer partition version is A324751. The integer partition version is A324756. The Heinz number version is A324758. An infinite version is A304360.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 15 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019
STATUS
approved