A331683 -id:A331683 - cp's OEIS frontend

A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root. It is an identity tree if no branch appears multiple times under the same root.

			The a(7) = 28 rooted trees:
  7,
  (16),
  (25),
  (1(15)),
  (34),
  (1(24)), (2(14)), (4(12)), (124),
  (1(1(14))),
  (3(13)),
  (2(23)),
  (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
  (1(1(1(13)))),
  (2(2(12))),
  (1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
  (1(1(1(1(12))))).
Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
The semi-identity tree version is A212804.
Not requiring local disjointness gives A300660.
The non-identity tree version is A316696.
This is the case of A331686 where all leaves are singletons.
Rooted identity trees are A004111.
Locally disjoint rooted identity trees are A316471.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Locally disjoint enriched identity p-trees are A331684.
Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,disjointQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(21)-a(23) from Robert Price, Sep 16 2018

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 4, 8, 17, 41, 103, 280, 793, 2330, 6979, 21291

Offset: 1

Gus Wiseman, Jan 31 2020

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (unequal) child of the same vertex. Lone-child-avoiding means there are no unary branchings. In an identity tree, all branches of any given vertex are distinct.

			The a(1) = 1 through a(5) = 17 trees:
  (1)  (2)   (3)       (4)            (5)
       (11)  (12)      (13)           (14)
             (111)     (22)           (23)
             ((1)(2))  (112)          (113)
                       (1111)         (122)
                       ((1)(3))       (1112)
                       ((2)(11))      (11111)
                       ((1)((1)(2)))  ((1)(4))
                                      ((2)(3))
                                      ((1)(22))
                                      ((3)(11))
                                      ((2)(111))
                                      ((1)((1)(3)))
                                      ((2)((1)(2)))
                                      ((11)((1)(2)))
                                      ((1)((2)(11)))
                                      ((1)((1)((1)(2))))

The non-identity version is A331678.
The case where the leaves are all singletons is A316694.
Identity trees are A004111.
Locally disjoint identity trees are A316471.
Locally disjoint enriched identity p-trees are A331684.
Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.

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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 133, 152, 172, 212, 214, 224, 256, 262, 266, 301, 304, 326, 344, 371, 424, 428, 448, 512, 524, 526, 532, 602, 608, 622, 652, 688, 742, 749, 766, 817, 848, 856, 886, 896, 917, 1007, 1024, 1048, 1052

Offset: 1

Gus Wiseman, Feb 04 2020

nonn

First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).
Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf branches of any given vertex are all distinct.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one, and all composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by Peter Munn and Gus Wiseman, Jun 24 2021]

			The sequence of all lone-child-avoiding rooted semi-identity trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
The sequence of terms together with their prime indices begins:
    1: {}                 224: {1,1,1,1,1,4}
    4: {1,1}              256: {1,1,1,1,1,1,1,1}
    8: {1,1,1}            262: {1,32}
   14: {1,4}              266: {1,4,8}
   16: {1,1,1,1}          301: {4,14}
   28: {1,1,4}            304: {1,1,1,1,8}
   32: {1,1,1,1,1}        326: {1,38}
   38: {1,8}              344: {1,1,1,14}
   56: {1,1,1,4}          371: {4,16}
   64: {1,1,1,1,1,1}      424: {1,1,1,16}
   76: {1,1,8}            428: {1,1,28}
   86: {1,14}             448: {1,1,1,1,1,1,4}
  106: {1,16}             512: {1,1,1,1,1,1,1,1,1}
  112: {1,1,1,1,4}        524: {1,1,32}
  128: {1,1,1,1,1,1,1}    526: {1,56}
  133: {4,8}              532: {1,1,4,8}
  152: {1,1,1,8}          602: {1,4,14}
  172: {1,1,14}           608: {1,1,1,1,1,8}
  212: {1,1,16}           622: {1,64}
  214: {1,28}             652: {1,1,38}

The non-semi case is {1}.
Not requiring lone-child-avoidance gives A306202.
The locally disjoint version is A331683.
These trees are counted by A331966.
The semi-lone-child-avoiding case is A331994.
Matula-Goebel numbers of rooted identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Semi-identity trees are counted by A306200.
Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.

csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n],{?(#>2&),?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],csiQ]

Intersection of A291636 and A306202.

A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 69, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 138, 144, 148, 152, 161, 162, 169, 172, 178, 184, 192, 196, 202, 206, 207, 208, 212, 214, 216, 224, 243

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))).
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one, two, and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Not requiring lone-child-avoidance gives A316495.
A superset of A320269.
The semi-identity tree case is A331681.
The non-semi version (i.e., not containing 2) is A331871.
These trees counted by vertices are A331872.
These trees counted by leaves are A331874.
Not requiring local disjointness gives A331935.
The identity tree case is A331937.
Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.

msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 55, 105, 200, 388, 754, 1483, 2923, 5807, 11575, 23190, 46608, 94043, 190287, 386214, 785831, 1602952, 3276845, 6712905, 13778079, 28330583, 58350582, 120370731, 248676129, 514459237, 1065696295, 2210302177, 4589599429, 9540623926

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Lone-child-avoiding means there are no unary branchings.

In a semi-identity tree, the non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        (o(o(oo)))  (oooo(oo))   (oooo(ooo))
                                                    ((oo)(ooo))  (ooooo(oo))
                                                    (o(o(ooo)))  ((oo)(oooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo)(ooo))
                                                                 (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 ((oo)(o(oo)))
                                                                 (o(o(o(oo))))

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The non-semi case is A000007.
Lone-child-avoiding rooted trees are A001678.
The locally disjoint case is A212804.
Not requiring lone-child-avoidance gives A306200.
Matula-Goebel numbers of these trees are A331965.
The semi-lone-child-avoiding version is A331993.
Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.

ssb[n_]:=If[n==1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[ssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssb[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020

Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020

A331912 Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 26, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 52, 53, 58, 59, 61, 64, 65, 67, 71, 73, 74, 79, 81, 83, 86, 87, 89, 91, 94, 97, 101, 103, 104, 107, 109, 111, 113, 116, 117, 121, 122, 125, 127, 128, 129, 131, 137

Offset: 1

Gus Wiseman, Feb 01 2020

nonn

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.

Conjecture: a(n)/A331784(n) -> 1 as n -> infinity.

			The sequence of terms together with their prime indices begins:
    1: {}              37: {12}              86: {1,14}
    2: {1}             39: {2,6}             87: {2,10}
    3: {2}             41: {13}              89: {24}
    4: {1,1}           43: {14}              91: {4,6}
    5: {3}             47: {15}              94: {1,15}
    7: {4}             49: {4,4}             97: {25}
    8: {1,1,1}         52: {1,1,6}          101: {26}
    9: {2,2}           53: {16}             103: {27}
   11: {5}             58: {1,10}           104: {1,1,1,6}
   13: {6}             59: {17}             107: {28}
   16: {1,1,1,1}       61: {18}             109: {29}
   17: {7}             64: {1,1,1,1,1,1}    111: {2,12}
   19: {8}             65: {3,6}            113: {30}
   23: {9}             67: {19}             116: {1,1,10}
   25: {3,3}           71: {20}             117: {2,2,6}
   26: {1,6}           73: {21}             121: {5,5}
   27: {2,2,2}         74: {1,12}           122: {1,18}
   29: {10}            79: {22}             125: {3,3,3}
   31: {11}            81: {2,2,2,2}        127: {31}
   32: {1,1,1,1,1}     83: {23}             128: {1,1,1,1,1,1,1}
For example, the prime indices of 117 are {2,2,6}, of which only 2 is already in the sequence, so 117 is in the sequence.

Gus Wiseman, Plot of A331912(n)/A331784(n) for n = 1..3729.

Contains all prime powers A000961.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with at most one prime index in S are A331784.
Numbers S with exactly one prime index in S are A331785.
Numbers S with exactly one distinct prime index in S are A331913.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.

aQ[n_]:=Length[Select[PrimePi/@First/@If[n==1,{},FactorInteger[n]],aQ]]<=1;
Select[Range[100],aQ]

A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047

Offset: 1

Gus Wiseman, Jan 25 2020

First differs from A320268 at a(11) = 45, A320268(11) = 44.

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.

			The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        ((oo)(oo))  (oooo(oo))   (oooo(ooo))
                                        (o(o(oo)))  (o(o(ooo)))  (ooooo(oo))
                                                    (o(oo)(oo))  ((ooo)(ooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo)(oo))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 (o((oo)(oo)))
                                                                 (o(o(o(oo))))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The enriched version is A316696.
The Matula-Goebel numbers of these trees are A331871.
The non-locally disjoint version is A001678.
These trees counted by number of leaves are A316697.
The semi-lone-child-avoiding version is A331872.
Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
strut[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]];
Table[Length[strut[n]],{n,10}]

A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 38, 48, 52, 56, 64, 74, 76, 86, 96, 104, 106, 112, 128, 148, 152, 172, 178, 192, 202, 208, 212, 214, 224, 256, 262, 296, 304, 326, 344, 356, 384, 404, 416, 424, 428, 446, 448, 478, 512, 524, 526, 592, 608, 622, 652

Offset: 1

Gus Wiseman, Jan 26 2020

nonn

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted semi-identity trees. A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct. Note that these conditions together imply that there is at most one non-leaf branch under any given vertex.
Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of all semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex, together with their Matula-Goebel numbers, begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  64: (oooooo)
  74: (o(oo(o)))
  76: (oo(ooo))
  86: (o(o(oo)))

Robert Israel, Table of n, a(n) for n = 1..4000

The enumeration of these trees by nodes is A324969 (essentially A000045).
The enumeration of these trees by leaves appears to be A090129(n + 1).
The (non-semi) lone-child-avoiding version is A331683.
Matula-Goebel numbers of rooted semi-identity trees are A306202.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
The set S of numbers with at most one prime index in S is A331784.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Cf. A000081, A000669, A001678, A007097, A061775, A196050, A291636, A316470, A316473, A331679, A331680, A331682, A331687, A331783, A331785, A331873.

N:= 1000: # for terms <= N
S:= {1,2}:
with(queue):
Q:= new(1,2):
while not empty(Q) do
  r:= dequeue(Q);
  p:= ithprime(r);
  newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
  S:= S union newS;
  for s in newS do enqueue(Q,s) od:
od:
sort(convert(S,list)); # Robert Israel, Feb 05 2020

uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,},{p,1}}/;uryQ[PrimePi[p]])|{{2,_}}];
Select[Range[100],uryQ]

Intersection of A306202 (semi-identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding). - Gus Wiseman, Jun 09 2020

A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 111, 113, 115, 119, 122, 127, 131, 133, 137, 139, 141, 142

Offset: 1

Gus Wiseman, Feb 01 2020

nonn

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.

Conjecture: A331912(n)/a(n) -> 1 as n -> infinity.

			The sequence of terms together with their prime indices begins:
    1: {}        43: {14}       91: {4,6}      141: {2,15}
    2: {1}       46: {1,9}      94: {1,15}     142: {1,20}
    3: {2}       47: {15}       95: {3,8}      143: {5,6}
    5: {3}       49: {4,4}      97: {25}       145: {3,10}
    7: {4}       53: {16}       98: {1,4,4}    147: {2,4,4}
   11: {5}       57: {2,8}     101: {26}       149: {35}
   13: {6}       58: {1,10}    103: {27}       151: {36}
   14: {1,4}     59: {17}      106: {1,16}     157: {37}
   17: {7}       61: {18}      107: {28}       158: {1,22}
   19: {8}       65: {3,6}     109: {29}       159: {2,16}
   21: {2,4}     67: {19}      111: {2,12}     161: {4,9}
   23: {9}       69: {2,9}     113: {30}       163: {38}
   26: {1,6}     71: {20}      115: {3,9}      167: {39}
   29: {10}      73: {21}      119: {4,7}      169: {6,6}
   31: {11}      74: {1,12}    122: {1,18}     173: {40}
   35: {3,4}     77: {4,5}     127: {31}       178: {1,24}
   37: {12}      79: {22}      131: {32}       179: {41}
   38: {1,8}     83: {23}      133: {4,8}      181: {42}
   39: {2,6}     87: {2,10}    137: {33}       182: {1,4,6}
   41: {13}      89: {24}      139: {34}       183: {2,18}
For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.

Gus Wiseman, Plot of A331912(n)/A331784(n) for n = 1..3729.

Contains all prime numbers A000040.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with exactly one prime index in S are A331785.
Numbers S with at most one distinct prime index in S are A331912.
Numbers S with exactly one distinct prime index in S are A331913.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aQ[n_]:=Length[Cases[primeMS[n],_?aQ]]<=1;
Select[Range[100],aQ]

A331875 Number of enriched identity p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 32, 79, 198, 522, 1368, 3716, 9992, 27612, 75692, 212045, 589478, 1668630, 4690792, 13387332, 37980664, 109098556, 311717768, 900846484, 2589449032, 7515759012, 21720369476, 63305262126, 183726039404, 537364221200, 1565570459800, 4592892152163

Offset: 1

Gus Wiseman, Jan 31 2020

nonn

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

			The a(1) = 1 through a(6) = 14 enriched p-trees:
  1  2  3     4        5           6
        (21)  (31)     (32)        (42)
              ((21)1)  (41)        (51)
                       ((21)2)     (321)
                       ((31)1)     ((21)3)
                       (((21)1)1)  ((31)2)
                                   ((32)1)
                                   (3(21))
                                   ((41)1)
                                   ((21)21)
                                   (((21)1)2)
                                   (((21)2)1)
                                   (((31)1)1)
                                   ((((21)1)1)1)

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

The orderless version is A300660.
The locally disjoint case is A331684.
Identity trees are A004111.
P-trees are A196545.
Enriched p-trees are A289501.
Cf. A000669, A141268, A306200, A316471, A331683, A331685, A331686, A331783.

eptrid[n_]:=Prepend[Select[Join@@Table[Tuples[eptrid/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&],n];
Table[Length[eptrid[n]],{n,10}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, sum(j=0, n\k, j!*binomial(v[k],j)*x^(k*j)) + O(x*x^n)), n)); v} \\ Andrew Howroyd, Feb 09 2020

Terms a(21) and beyond from Andrew Howroyd, Feb 09 2020

cp's OEIS Frontend

A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

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A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

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A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

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A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

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A331912 Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.

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A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

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A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

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