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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

Table of n, a(n) for n=2..71.

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.

Cf. A000111, A002846, A005121, A292504, A318812, A318813, A318847, A318848, A318849, A330475, A330666, A330935.

Sequence in context: A107338 A118123 A181743 * A174737 A131756 A212620

Adjacent sequences:  A330724 A330725 A330726 * A330728 A330729 A330730

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jan 04 2020

STATUS

approved

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Last modified June 16 16:09 EDT 2021. Contains 345063 sequences. (Running on oeis4.)