OFFSET
1,6
COMMENTS
For n > 1, the first appearance of n is 2 * A080696(n - 1).
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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
EXAMPLE
The multiset of multisets with MM-number 1508 is {{},{},{1,2},{1,3}}, which has the 3 connected components {{}}, {{}}, and {{1,2},{1,3}}, two of which are distinct, so a(1508) = 2.
The multiset of multisets with MM-number 12818 is {{},{1,2},{4},{1,3}}, which has the 3 connected components {{}}, {{1,2},{1,3}}, and {{4}}, so a(12818) = 3.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Length[Union[zsm[primeMS[n]]]], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 20 2019
STATUS
approved