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A324741
Number of subsets of {1...n} containing no prime indices of the elements.
19
1, 2, 3, 5, 8, 13, 19, 30, 54, 96, 156, 248, 440, 688, 1120, 1864, 3664, 5856, 11232, 16896, 31296, 53952, 91008, 137472, 270528, 516720, 863088, 1710816, 3173856, 4836672, 9329472, 14897376, 29788128, 52256448, 88429248, 166037184, 331648704, 497685888, 829449600
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
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.
Sequence in context: A326469 A326594 A045842 * A200462 A088795 A156145
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 15 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019
STATUS
approved