A005804 -id:A005804 - cp's OEIS frontend

A316653 Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.

1, 1, 6, 58, 774, 13171, 272700, 6655962, 187172762, 5959665653, 211947272186, 8327259067439, 358211528524432, 16744766791743136, 845195057333580332, 45814333121920927067, 2654330505021077873594, 163687811930206581162063, 10705203621191765328300832

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

			The a(3) = 6 trees are (1(12)), (2(12)), (1(23)), (2(13)), (3(12)), (123).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A141268, A292504, A300660.

Cf. A316651, A316652, A316654, A316655, A316656.

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]]]];
gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]

\\ here R(n,2) is A031148.
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, WeighT(concat(v,[0]))[n])); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018

A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 4, 3, 1, 0, 9, 0, 1, 6, 26, 0, 36, 0, 16, 10, 1, 0, 92, 21, 1, 197, 25, 0, 100, 0, 236, 15, 1, 53, 474

Offset: 1

Gus Wiseman, Jul 09 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of sets of trees begins:
   1:
   2: 1
   3:
   4: (12)
   5:
   6: (1(12))
   7:
   8: (1(23)), (2(13)), (3(12)), (123)
   9: (1(2(12))), (2(1(12))), (12(12))
  10: (1(1(12)))
  11:
  12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12))

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653. A316654, A316655.

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]]]];
gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,30}]

a(prime(n>1)) = 0.

a(2^n) = A000311(n).

A330467 Number of series-reduced rooted trees whose leaves are multisets whose multiset union is a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 4, 18, 154, 1614, 23733, 396190, 8066984, 183930948, 4811382339, 138718632336, 4451963556127, 155416836338920, 5920554613563841, 242873491536944706, 10725017764009207613, 505671090907469848248, 25415190929321149684700, 1354279188424092012064226

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Also the number of different colorings of phylogenetic trees with n labels using strongly normal multisets of colors. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			The a(3) = 18 trees:
  {1,1,1}          {1,1,2}          {1,2,3}
  {{1},{1,1}}      {{1},{1,2}}      {{1},{2,3}}
  {{1},{1},{1}}    {{2},{1,1}}      {{2},{1,3}}
  {{1},{{1},{1}}}  {{1},{1},{2}}    {{3},{1,2}}
                   {{1},{{1},{2}}}  {{1},{2},{3}}
                   {{2},{{1},{1}}}  {{1},{{2},{3}}}
                                    {{2},{{1},{3}}}
                                    {{3},{{1},{2}}}

The singleton-reduced version is A316652.
The unlabeled version is A330465.
Not requiring weakly decreasing multiplicities gives A330469.
The case where the leaves are sets is A330625.
Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A318812, A318849, A319312, A330471, A330475.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]],Length[s]],{s,Split[c]}],{c,Select[mps[m],Length[#]>1&]}];
Table[Sum[amemo[m],{m,strnorm[n]}],{n,0,5}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sExp(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n ) + polcoef(p, n)); 1 + x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 28 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 28 2020

A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 24, 250, 3744, 73408, 1768088, 50468854, 1664844040, 62304622320, 2607765903568, 120696071556230, 6120415124163512, 337440974546042416, 20096905939846645064, 1285779618228281270718, 87947859243850506008984, 6404472598196204610148232

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

Also the number of different colorings of phylogenetic trees with n labels using a multiset of colors covering an initial interval of positive integers. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			The a(3) = 24 trees:
  (123)          (122)          (112)          (111)
  ((1)(23))      ((1)(22))      ((1)(12))      ((1)(11))
  ((2)(13))      ((2)(12))      ((2)(11))      ((1)(1)(1))
  ((3)(12))      ((1)(2)(2))    ((1)(1)(2))    ((1)((1)(1)))
  ((1)(2)(3))    ((1)((2)(2)))  ((1)((1)(2)))
  ((1)((2)(3)))  ((2)((1)(2)))  ((2)((1)(1)))
  ((2)((1)(3)))
  ((3)((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 0..200

The singleton-reduced version is A316651.
The unlabeled version is A330465.
The strongly normal case is A330467.
The case when leaves are sets is A330764.
Row sums of A330762.
Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A316652, A318812, A318849, A319312, A330625.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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]]]];
multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]],Length[s]],{s,Split[c]}],{c,Select[mps[m],Length[#]>1&]}];
Table[Sum[amemo[m],{m,allnorm[n]}],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
seq(n)={concat([1], sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 29 2019

Terms a(9) and beyond from Andrew Howroyd, Dec 29 2019

A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 6, 43, 440, 7158, 151414

Offset: 0

Gus Wiseman, Dec 30 2019

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 finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

			The a(2) = 2 and a(3) = 6 multisystems:
  {1,1}  {{1},{1,1}}
  {1,2}  {{1},{1,2}}
         {{1},{2,3}}
         {{2},{1,1}}
         {{2},{1,3}}
         {{3},{1,2}}
The a(4) = 43 multisystems (commas and outer brackets elided):
  {{1}}{{1}{11}} {{1}}{{1}{12}} {{1}}{{1}{22}} {{1}}{{1}{23}} {{1}}{{2}{34}}
  {{11}}{{1}{1}} {{11}}{{1}{2}} {{11}}{{2}{2}} {{11}}{{2}{3}} {{12}}{{3}{4}}
                 {{1}}{{2}{11}} {{1}}{{2}{12}} {{1}}{{2}{13}} {{1}}{{3}{24}}
                 {{12}}{{1}{1}} {{12}}{{1}{2}} {{12}}{{1}{3}} {{13}}{{2}{4}}
                 {{2}}{{1}{11}} {{2}}{{1}{12}} {{1}}{{3}{12}} {{1}}{{4}{23}}
                                {{2}}{{2}{11}} {{13}}{{1}{2}} {{14}}{{2}{3}}
                                {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{1}{34}}
                                               {{2}}{{3}{11}} {{2}}{{3}{14}}
                                               {{23}}{{1}{1}} {{23}}{{1}{4}}
                                               {{3}}{{1}{12}} {{2}}{{4}{13}}
                                               {{3}}{{2}{11}} {{24}}{{1}{3}}
                                                              {{3}}{{1}{24}}
                                                              {{3}}{{2}{14}}
                                                              {{3}}{{4}{12}}
                                                              {{34}}{{1}{2}}
                                                              {{4}}{{1}{23}}
                                                              {{4}}{{2}{13}}
                                                              {{4}}{{3}{12}}

The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330475.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The (weakly) normal version is A330676.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of strongly normal multisets are A035310.
Series-reduced rooted trees with strongly normal leaves are A316652.
Cf. A000311, A000669, A001055, A001678, A005121, A005804, A316651, A318812, A330467, A330474, A330625, A330628, A330664, A330677, A330679.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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

A316654 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 1, 5, 39, 387, 4960, 74088, 1312716, 26239484, 595023510, 14908285892, 412903136867, 12448252189622, 407804188400373, 14380454869464352, 544428684832123828, 21991444994187529639, 945234507638271696504, 43042162953650721470752, 2071216980365429970912347

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

			The a(3) = 5 trees are (1(12)), (1(23)), (2(13)), (3(12)), (123).

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A141268, A181821, A292504, A300660, A304660.

Cf. A316651, A316652, A316653, A316655, A316656.

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]]]];
gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n]=polcoef(sWeighT(x*Ser(v[1..n])), n)); x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(12)) \\ Andrew Howroyd, Jan 22 2021

Terms a(9) and beyond from Andrew Howroyd, Jan 22 2021

A318849 Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 11, 8, 27, 20, 30, 38, 96, 74, 114, 58, 308, 234, 1052, 176, 509, 278, 3648, 374, 600, 1076, 1760, 814, 13003, 1306, 47006, 612, 2226, 4200, 3094, 2914, 172605, 16588, 9814, 2168, 640662, 6998, 2402388, 3698, 11496, 65936, 9082538, 4914, 17996

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset 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}.

A tree-partition of m is either m itself or a multiset of tree-partitions, one of each part of a multiset partition of m with at least two parts.

			The a(7) = 11 orderless tree-partitions of {1,1,1,1}:
  (1111)
  ((1)(111))
  ((11)(11))
  ((1)(1)(11))
  ((1)((1)(11)))
  ((11)((1)(1)))
  ((1)(1)(1)(1))
  ((1)((1)(1)(1)))
  ((1)(1)((1)(1)))
  ((1)((1)((1)(1))))
  (((1)(1))((1)(1)))

Cf. A000311, A001055, A196545, A292504, A292505, A305936, A316655, A318762, A318812, A318813, A318847.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
olmsptrees[m_]:=Prepend[Union@@Table[Sort/@Tuples[olmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[olmsptrees[nrmptn[n]]],{n,15}]

a(n) = A292504(A181821(n)).
a(prime(n)) = A141268(n).
a(2^n) = A005804(n).

More terms from Jinyuan Wang, Jun 26 2020

A320169 Number of balanced enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 3, 6, 9, 20, 31, 70, 114, 243, 415, 961, 1603, 3564, 6559, 14913, 26630, 60037, 110160, 248859, 458445, 1001190, 1882350, 4220358, 7765303, 16822107, 32307240, 70081784, 133716083, 291788153, 561823990, 1230204229, 2396185727, 5176454708, 10220127290

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

An enriched p-tree of weight n is either the number n itself or a finite sequence of enriched p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(6) = 20 balanced enriched p-trees:
  1  2     3      4           5            6
     (11)  (21)   (22)        (32)         (33)
           (111)  (31)        (41)         (42)
                  (211)       (221)        (51)
                  (1111)      (311)        (222)
                  ((11)(11))  (2111)       (321)
                              (11111)      (411)
                              ((21)(11))   (2211)
                              ((111)(11))  (3111)
                                           (21111)
                                           (111111)
                                           ((21)(21))
                                           ((22)(11))
                                           ((31)(11))
                                           ((111)(21))
                                           ((21)(111))
                                           ((211)(11))
                                           ((111)(111))
                                           ((1111)(11))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

Cf. A316624, A320154, A320155, A320160, A320179.

eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p],{p,Rest[IntegerPartitions[n]]}],n];
Table[Length[Select[eptrs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,12}]

seq(n)={my(p=x/(1-x) + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 26 2018

A330624 Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.

Original entry on oeis.org

1, 1, 3, 10, 61, 410, 3630

Offset: 0

Gus Wiseman, Dec 25 2019

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 10 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}

The version with multisets as leaves is A330465.
The singleton-reduced case is A330626.
A labeled version is A330625 (strongly normal).
The case with all atoms distinct is A141268.
The case where all leaves are singletons is A330470.
Cf. A000669, A004111, A005804, A007716, A273873, A300660, A320296, A330628, A330668, A330677.

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 27 2019

nonn

First differs from A317145 at a(32) = 5, A317145(32) = 4.
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.
Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

			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}}}}

The last nonzero term in row n of A330667 is a(n).
The chain version is A317145.
The non-maximal version is A318812.
Unlabeled versions are A330664 and A330663.
Other labeled versions are A330675 (strongly normal) and A330676 (normal).
Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.

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

a(2^n) = A000111(n - 1).

a(product of n distinct primes) = A006472(n).

cp's OEIS Frontend

A316653 Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.

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A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

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A330467 Number of series-reduced rooted trees whose leaves are multisets whose multiset union is a strongly normal multiset of size n.

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A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.

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A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

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A316654 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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A318849 Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.

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A320169 Number of balanced enriched p-trees of weight n.

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