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A322306
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Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).
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6
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0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 4, 2, 2, 3, 2, 1, 3, 1, 2, 5, 1, 3, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 4, 2, 1, 4, 2, 2, 3
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OFFSET
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1,6
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. A positive integer is connected if its prime indices are connected (see A305078).
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LINKS
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EXAMPLE
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The a(1365) = 12 divisors are 3, 5, 7, 13, 21, 39, 65, 91, 195, 273, 455, 1365. These correspond to the following connected submultisets of {{1},{2},{1,1},{1,2}}.
3: {{1}}
5: {{2}}
7: {{1,1}}
13: {{1,2}}
21: {{1},{1,1}}
39: {{1},{1,2}}
65: {{2},{1,2}}
91: {{1,1},{1,2}}
195: {{1},{2},{1,2}}
273: {{1},{1,1},{1,2}}
455: {{2},{1,1},{1,2}}
1365: {{1},{2},{1,1},{1,2}}
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MATHEMATICA
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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[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[Union[Subsets[primeMS[n]]], Length[zsm[#]]==1&]], {n, 50}]
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CROSSREFS
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Cf. A003963, A054921, A112798, A286518, A290103, A301957, A302242, A304714, A304716, A305078, A316556, A322307.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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