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 A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n. 6
 1, 1, 1, 1, 1, 2, 2, 3, 7, 5, 5, 11, 16, 16, 27, 18, 61, 62, 272, 45, 123, 61, 1385, 105, 152, 272, 501, 211, 7936, 362 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS 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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}. LINKS Table of n, a(n) for n=1..30. FORMULA a(2^n) = A006472(n). a(prime(n)) = A000111(n - 1). EXAMPLE The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided): 11 {1}{12} {1}{23} {{1}}{{1}{22}} {{1}}{{1}{12}} {{1}}{{1}{23}} {2}{11} {2}{13} {{11}}{{2}{2}} {{11}}{{1}{2}} {{11}}{{2}{3}} {3}{12} {{1}}{{2}{12}} {{1}}{{2}{11}} {{1}}{{2}{13}} {{12}}{{1}{2}} {{12}}{{1}{1}} {{12}}{{1}{3}} {{2}}{{1}{12}} {{2}}{{1}{11}} {{1}}{{3}{12}} {{2}}{{2}{11}} {{13}}{{1}{2}} {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{3}{11}} {{23}}{{1}{1}} {{3}}{{1}{12}} {{3}}{{2}{11}} MATHEMATICA nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[Reverse[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

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Last modified April 22 08:40 EDT 2024. Contains 371893 sequences. (Running on oeis4.)