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A328514
MM-numbers of connected sets of sets.
15
1, 2, 3, 5, 11, 13, 17, 29, 31, 39, 41, 43, 47, 59, 65, 67, 73, 79, 83, 87, 101, 109, 113, 127, 129, 137, 139, 149, 157, 163, 167, 179, 181, 191, 195, 199, 211, 233, 235, 237, 241, 257, 269, 271, 277, 283, 293, 303, 313, 317, 319, 331, 339, 347, 349, 353, 365
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. 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}}.
FORMULA
Intersection of A302494 and A305078.
EXAMPLE
The sequence all connected set of sets together with their MM-numbers begins:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
11: {{3}}
13: {{1,2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
39: {{1},{1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
59: {{7}}
65: {{2},{1,2}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
87: {{1},{1,3}}
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]]]]]]]]];
Select[Range[1000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]
CROSSREFS
The not-necessarily-connected case is A302494.
BII-numbers of connected set-systems are A326749.
MM-numbers of connected sets of multisets are A328513.
Sequence in context: A127046 A127051 A127045 * A302491 A141830 A127048
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 20 2019
STATUS
approved