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A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n. 5
1, 1, 1, 1, 2, 1, 3, 2, 1, 3, 1, 7, 7, 1, 5, 5, 1, 5, 9, 5, 1, 9, 11, 1, 9, 28, 36, 16, 1, 10, 24, 16, 1, 14, 38, 27, 1, 13, 18, 1, 13, 69, 160, 164, 61, 1, 24, 79, 62, 1, 20, 160, 580, 1022, 855, 272, 1, 19, 59, 45, 1, 27, 138, 232, 123, 1, 17, 77, 121, 61 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,5
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
FORMULA
T(2^n,k) = A008826(n,k).
EXAMPLE
Triangle begins:
{}
1
1
1 1
1 2
1 3 2
1 3
1 7 7
1 5 5
1 5 9 5
1 9 11
1 9 28 36 16
1 10 24 16
1 14 38 27
1 13 18
1 13 69 160 164 61
1 24 79 62
For example, row n = 12 counts the following multisystems:
{1,1,2,3} {{1},{1,2,3}} {{{1}},{{1},{2,3}}}
{{1,1},{2,3}} {{{1,1}},{{2},{3}}}
{{1,2},{1,3}} {{{1}},{{2},{1,3}}}
{{2},{1,1,3}} {{{1,2}},{{1},{3}}}
{{3},{1,1,2}} {{{1}},{{3},{1,2}}}
{{1},{1},{2,3}} {{{1,3}},{{1},{2}}}
{{1},{2},{1,3}} {{{2}},{{1},{1,3}}}
{{1},{3},{1,2}} {{{2}},{{3},{1,1}}}
{{2},{3},{1,1}} {{{2,3}},{{1},{1}}}
{{{3}},{{1},{1,2}}}
{{{3}},{{2},{1,1}}}
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<Length[#]<Length[m]&]}], m];
Table[Length[Select[totm[nrmptn[n]], Depth[#]==k&]], {n, 2, 10}, {k, 2, Length[nrmptn[n]]}]
CROSSREFS
Row sums are A318846.
Final terms in each row are A330728.
Row prime(n) is row n of A330784.
Row 2^n is row n of A008826.
Row n is row A181821(n) of A330667.
Column k = 3 is A318284(n) - 2 for n > 2.
Sequence in context: A107338 A118123 A181743 * A174737 A131756 A212620
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 04 2020
STATUS
approved

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Last modified July 19 17:03 EDT 2024. Contains 374410 sequences. (Running on oeis4.)