|
|
A324736
|
|
Number of subsets of {1...n} containing all prime indices of the elements.
|
|
21
|
|
|
1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(0) = 1 through a(6) = 15 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{1,2} {1,2} {1,2} {1,2} {1,2}
{1,2,3} {1,4} {1,4} {1,4}
{1,2,3} {1,2,3} {1,2,3}
{1,2,4} {1,2,4} {1,2,4}
{1,2,3,4} {1,2,3,4} {1,2,6}
{1,2,3,5} {1,2,3,4}
{1,2,3,4,5} {1,2,3,5}
{1,2,3,6}
{1,2,4,6}
{1,2,3,4,5}
{1,2,3,4,6}
{1,2,3,5,6}
{1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
1: {}
2: {1}
4: {1,1}
7: {4}
8: {1,1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
17: {7}
18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.
|
|
MATHEMATICA
|
Table[Length[Select[Subsets[Range[n]], SubsetQ[#, PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#, 1]]&]], {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(!bitnegimply(p[k], b), t+=if(bittest(d, k), self()(k+1, b+(1<<k)), t)); t))(1, 0)} \\ Andrew Howroyd, Aug 15 2019
|
|
CROSSREFS
|
The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|