A292505 -id:A292505 - cp's OEIS frontend

A292504 Number of orderless tree-factorizations of n.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 11, 1, 6, 1, 6, 2, 2, 1, 20, 2, 2, 4, 6, 1, 8, 1, 30, 2, 2, 2, 27, 1, 2, 2, 20, 1, 8, 1, 6, 6, 2, 1, 74, 2, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 38, 1, 2, 6, 96, 2, 8, 1, 6, 2, 8, 1, 114, 1, 2, 6, 6, 2, 8, 1, 74, 11, 2, 1

Offset: 1

Gus Wiseman, Sep 17 2017

nonn

A factorization of n is a finite multiset of positive integers greater than 1 with product n. An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless tree-factorizations, one of each factor in a factorization of n.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(16)=11 orderless tree-factorizations are: 16, (28), (2(24)), (2(2(22))), (2(222)), (44), (4(22)), ((22)(22)), (224), (22(22)), (2222).

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Records and indices of records.

Cf. A000311, A001055, A050336, A141268, A281118, A292505.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[oltfacs[n]],{n,83}]

seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(e+w[k]-1, e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A141268(n) for prime p. - Andrew Howroyd, Nov 18 2018

A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

Original entry on oeis.org

1, 1, 2, 7, 39, 236, 1836, 16123, 162008, 1802945, 22012335, 291290460, 4144907830, 62986968311, 1016584428612, 17344929138791, 311618472138440, 5875109147135658, 115894178676866576, 2385755803919949337, 51133201045333895149, 1138659323863266945177, 26296042933904490636133

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

			Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:
  (1111)      (1112)      (1122)      (1123)      (1234)
  (1(111))    (1(112))    (1(122))    (1(123))    (1(234))
  (11(11))    (11(12))    (11(22))    (11(23))    (12(34))
  ((11)(11))  (12(11))    (12(12))    (12(13))    ((12)(34))
  (1(1(11)))  (2(111))    ((11)(22))  (2(113))    (1(2(34)))
              ((11)(12))  (1(1(22)))  (23(11))
              (1(1(12)))  ((12)(12))  ((11)(23))
              (1(2(11)))  (1(2(12)))  (1(1(23)))
              (2(1(11)))              ((12)(13))
                                      (1(2(13)))
                                      (2(1(13)))
                                      (2(3(11)))

The case with all atoms equal or all atoms different is A000669.
Not requiring singleton-reduction gives A330465.
Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).
The case where the leaves are sets is A330626.
Row sums of A339645.
Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.

\\ 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( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 11 2020

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

A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 5, 8, 4, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 1, 0, 1, 0, 1, 0, 1, 5, 5, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 04 2020

This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.

			Triangle begins:
   1:          16: 0 1 3 2    31: 1            46: 0 1
   2: 1        17: 1          32: 0 1 5 8 4    47: 1
   3: 1        18: 0 1 2      33: 0 1          48: 0 1 10 23 15
   4: 0 1      19: 1          34: 0 1          49: 0 1
   5: 1        20: 0 1 2      35: 0 1          50: 0 1 2
   6: 0 1      21: 0 1        36: 0 1 7 7      51: 0 1
   7: 1        22: 0 1        37: 1            52: 0 1 2
   8: 0 1 1    23: 1          38: 0 1          53: 1
   9: 0 1      24: 0 1 5 5    39: 0 1          54: 0 1 5 5
  10: 0 1      25: 0 1        40: 0 1 5 5      55: 0 1
  11: 1        26: 0 1        41: 1            56: 0 1 5 5
  12: 0 1 2    27: 0 1 1      42: 0 1 3        57: 0 1
  13: 1        28: 0 1 2      43: 1            58: 0 1
  14: 0 1      29: 1          44: 0 1 2        59: 1
  15: 0 1      30: 0 1 3      45: 0 1 2        60: 0 1 9 11
Row n = 48 counts the following chains (minimum and maximum not shown):
  ()  (6*8)      (2*3*8)->(6*8)       (2*2*2*6)->(2*4*6)->(6*8)
      (2*24)     (2*4*6)->(6*8)       (2*2*3*4)->(2*3*8)->(6*8)
      (3*16)     (2*3*8)->(2*24)      (2*2*3*4)->(2*4*6)->(6*8)
      (4*12)     (2*3*8)->(3*16)      (2*2*2*6)->(2*4*6)->(2*24)
      (2*3*8)    (2*4*6)->(2*24)      (2*2*2*6)->(2*4*6)->(4*12)
      (2*4*6)    (2*4*6)->(4*12)      (2*2*3*4)->(2*3*8)->(2*24)
      (3*4*4)    (3*4*4)->(3*16)      (2*2*3*4)->(2*3*8)->(3*16)
      (2*2*12)   (3*4*4)->(4*12)      (2*2*3*4)->(2*4*6)->(2*24)
      (2*2*2*6)  (2*2*12)->(2*24)     (2*2*3*4)->(2*4*6)->(4*12)
      (2*2*3*4)  (2*2*12)->(4*12)     (2*2*3*4)->(3*4*4)->(3*16)
                 (2*2*2*6)->(6*8)     (2*2*3*4)->(3*4*4)->(4*12)
                 (2*2*3*4)->(6*8)     (2*2*2*6)->(2*2*12)->(2*24)
                 (2*2*2*6)->(2*24)    (2*2*2*6)->(2*2*12)->(4*12)
                 (2*2*2*6)->(4*12)    (2*2*3*4)->(2*2*12)->(2*24)
                 (2*2*3*4)->(2*24)    (2*2*3*4)->(2*2*12)->(4*12)
                 (2*2*3*4)->(3*16)
                 (2*2*3*4)->(4*12)
                 (2*2*2*6)->(2*4*6)
                 (2*2*3*4)->(2*3*8)
                 (2*2*3*4)->(2*4*6)
                 (2*2*3*4)->(3*4*4)
                 (2*2*2*6)->(2*2*12)
                 (2*2*3*4)->(2*2*12)

Row lengths are A001222.
Row sums are A317176.
Column k = 1 is A010051.
Column k = 2 is A066247.
Column k = 3 is A330936.
Final terms of each row are A317145.
The version for set partitions is A008826, with row sums A005121.
The version for integer partitions is A330785, with row sums A213427.
Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upfacs[y],y],{y,facs[n]}],{n},First[facs[n]]],Length[#]==k-1&]],{n,100},{k,PrimeOmega[n]}]

T(2^n,k) = A330785(n,k).

T(n,1) + T(n,2) = 1.

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

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).

A277130 Number of planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 6, 2, 3, 1, 14, 1, 3, 3, 24, 1, 14, 1, 14, 3, 3, 1, 78, 2, 3, 6, 14, 1, 25, 1, 112, 3, 3, 3, 110, 1, 3, 3, 78, 1, 25, 1, 14, 14, 3, 1, 464, 2, 14, 3, 14, 1, 78, 3, 78, 3, 3, 1, 206, 1, 3, 14, 568, 3, 25, 1, 14, 3, 25, 1, 850, 1, 3, 14, 14

Offset: 1

Michel Marcus, Oct 01 2016

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. - Gus Wiseman, Sep 11 2018

			From _Gus Wiseman_, Sep 11 2018: (Start)
The a(12) = 14 planar branching factorizations:
  12,
  (2*6), (3*4), (4*3), (6*2), (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
(End)

Daniel Mondot, Table of n, a(n) for n = 1..9999
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 15.

Cf. A277120.
Cf. A000108, A001055, A074206, A118376, A281113, A319122, A319123.
Cf. A277130, A281118, A281119, A292504, A292505, A295279, A295281, A319136, A319137, A319138.

#include 
#include 
#include 
#define MAX 10000
/* Number of planar branching factorizations of n. */
#define lu unsigned long
lu nbr[MAX]; /* number of branching */
lu a, b, d, e; /* temporary variables */
lu n; lu m, p; // factors of n
lu x; // square root of n
void main(unsigned argc, char *argv[])
{
  memset(nbr, 0, MAX*sizeof(lu));
  for (b=0, n=1; nDaniel Mondot, May 19 2017 */

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[otfs[n]],{n,20}] (* Gus Wiseman, Sep 11 2018 *)

a(prime^n) = A118376(n). a(product of n distinct primes) = A319122(n). - Gus Wiseman, Sep 11 2018

Terms a(65) and beyond from Daniel Mondot, May 19 2017

A295281 Number of complete strict tree-factorizations of n > 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 4, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 0, 1, 4, 1, 1, 1, 4, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 0, 1, 1, 9, 1, 1, 1, 1, 1, 9, 1

Offset: 2

Gus Wiseman, Nov 19 2017

nonn

A strict tree-factorization (see A295279 for definition) is complete if its leaves are all prime numbers.
From Andrew Howroyd, Nov 18 2018: (Start)
a(n) depends only on the prime signature of n.
This sequence is very similar but not identical to the number of complete orderless identity tree-factorizations of n. The first difference is at n=900 (square of three primes). Here a(n) = 191 whereas the other sequence would have 197. (End)

			The a(72) = 6 complete strict tree-factorizations are: 2*3*(2*(2*3)), 2*(2*3*(2*3)), 2*(2*(3*(2*3))), 2*(3*(2*(2*3))), 3*(2*(2*(2*3))), (2*3)*(2*(2*3)).

Andrew Howroyd, Table of n, a(n) for n = 2..10000

Cf. A000311, A025475, A085971, A273873, A281113 A281118, A281119, A292505, A295279.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sftc[n_]:=Prepend[Join@@Function[fac,Tuples[sftc/@fac]]/@Select[postfacs[n],And[Length[#]>1,UnsameQ@@#]&],n];
Table[Length[Select[sftc[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,100}]

seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, v[j]+=w[k]*v[j/k])); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(product of n distinct primes) = A000311(n).

Positions of zeros are proper prime powers A025475. Positions of nonzero entries are A085971.

A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

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

Offset: 1

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 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}.

			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}}

The version with distinct atoms is A006472.
The non-maximal version is A318846.
A tree version is A318848, with orderless version A318849.
The unlabeled version is A330664.
Final terms in each row of A330727.
See also A330675 (strongly normal), A330676 (normal), and A330726 (partition).
Cf. A000111, A001055, A005121, A292504, A292505, A317145, A330665, A330666.

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

a(2^n) = A006472(n).

a(prime(n)) = A000111(n - 1).

cp's OEIS Frontend

A292504 Number of orderless tree-factorizations of n.

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A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size 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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A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

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

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A277130 Number of planar branching factorizations of n.

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