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A330665
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Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.
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9
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 16, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 16, 1, 3, 1, 2, 1, 3, 1, 27, 1, 1, 2, 2, 1, 3, 1, 16, 2, 1, 1, 11, 1
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OFFSET
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1,12
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COMMENTS
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A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
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.
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LINKS
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FORMULA
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a(product of n distinct primes) = A006472(n).
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EXAMPLE
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The a(n) multisystems for n = 2, 6, 12, 24, 48:
{1} {1,2} {{1},{1,2}} {{{1}},{{1},{1,2}}} {{{{1}}},{{{1}},{{1},{1,2}}}}
{{2},{1,1}} {{{1,1}},{{1},{2}}} {{{{1}}},{{{1,1}},{{1},{2}}}}
{{{1}},{{2},{1,1}}} {{{{1},{1}}},{{{1}},{{1,2}}}}
{{{1,2}},{{1},{1}}} {{{{1},{1,1}}},{{{1}},{{2}}}}
{{{2}},{{1},{1,1}}} {{{{1,1}}},{{{1}},{{1},{2}}}}
{{{{1}}},{{{1}},{{2},{1,1}}}}
{{{{1}}},{{{1,2}},{{1},{1}}}}
{{{{1},{1}}},{{{2}},{{1,1}}}}
{{{{1},{1,2}}},{{{1}},{{1}}}}
{{{{1,1}}},{{{2}},{{1},{1}}}}
{{{{1}}},{{{2}},{{1},{1,1}}}}
{{{{1},{2}}},{{{1}},{{1,1}}}}
{{{{1,2}}},{{{1}},{{1},{1}}}}
{{{{2}}},{{{1}},{{1},{1,1}}}}
{{{{2}}},{{{1,1}},{{1},{1}}}}
{{{{2},{1,1}}},{{{1}},{{1}}}}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p], {p, Select[mps[m], 1<Length[#]<Length[m]&]}], m];
Table[Length[Select[totm[primeMS[n]], Length[#]<=1||Depth[#]==PrimeOmega[n]&]], {n, 100}]
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CROSSREFS
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The last nonzero term in row n of A330667 is a(n).
The non-maximal version is A318812.
Other labeled versions are A330675 (strongly normal) and A330676 (normal).
Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.
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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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