OFFSET
1,2
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.
By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
Heinz numbers of the partitions counted by A239312.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
5: {3}
6: {1,2}
7: {4}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
19: {8}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
26: {1,6}
29: {10}
30: {1,2,3}
MAPLE
filter:= proc(n) uses numtheory, GraphTheory; local B, S, F, D, E, G, t, d;
F:= ifactors(n)[2];
F:= map(t -> [pi(t[1]), t[2]], F);
D:= `union`(seq(divisors(t[1]), t = F));
F:= map(proc(t) local i; seq([t[1], i], i=1..t[2]) end proc, F);
if nops(D) < nops(F) then return false fi;
E:= {seq(seq({t, d}, d=divisors(t[1])), t = F)};
S:= map(t -> convert(t, name), [op(F), op(D)]);
E:= map(e -> map(convert, e, name), E);
G:= Graph(S, E);
B:= BipartiteMatching(G);
B[1] = nops(F);
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 15 2024
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Tuples[Divisors/@prix[#]], UnsameQ@@#&]!={}&]
CROSSREFS
Positions of nonzero terms in A355739.
Complement of A355740.
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 15 2023
STATUS
approved