Displaying 1-10 of 11 results found.
One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.
+10
20
1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 152, 172, 212, 214, 224, 256, 262, 304, 326, 344, 424, 428, 448, 512, 524, 526, 608, 622, 652, 688, 766, 848, 856, 886, 896, 1024, 1048, 1052, 1154, 1216, 1226, 1244, 1304, 1376, 1438, 1532, 1696
COMMENTS
Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if there are no unary branchings. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. 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.
EXAMPLE
The sequence of all 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
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)
152: (ooo(ooo))
172: (oo(o(oo)))
212: (oo(oooo))
214: (o(oo(oo)))
224: (ooooo(oo))
MAPLE
N:= 10^4: # for terms <= N
S:= {1}:
with(queue):
Q:= new(1):
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:
MATHEMATICA
uryQ[n_]:=n==1||MatchQ[FactorInteger[n], ({{2, _}, {p_, 1}}/; uryQ[PrimePi[p]])|({{2, k_}}/; k>1)];
Select[Range[100], uryQ]
CROSSREFS
These trees counted by number of vertices are A212804. The semi-lone-child-avoiding version is A331681. The non-semi-identity version is A331871. Lone-child-avoiding rooted trees are counted by A001678. Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636. Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202. Locally disjoint rooted trees are counted by A316473. Matula-Goebel numbers of locally disjoint rooted trees are A316495. Lone-child-avoiding locally disjoint rooted trees by leaves are A316697. Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.
Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.
+10
16
1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870
COMMENTS
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.
EXAMPLE
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))).
MATHEMATICA
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}]
CROSSREFS
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. Locally disjoint rooted identity trees are A316471. Lone-child-avoiding locally disjoint rooted trees are A331680. Locally disjoint enriched identity p-trees are A331684.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Feb 06 2020
Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.
+10
15
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
COMMENTS
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]
EXAMPLE
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}
MATHEMATICA
csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n], {_?(#>2&), _?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100], csiQ]
CROSSREFS
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.
Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.
+10
14
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
COMMENTS
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.
EXAMPLE
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))
MATHEMATICA
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]
CROSSREFS
Not requiring lone-child-avoidance gives A316495. 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. Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.
Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.
+10
12
1, 2, 3, 8, 16, 48, 116, 341, 928, 2753, 7996, 24254, 73325, 226471, 702122
COMMENTS
A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.
EXAMPLE
The a(1) = 1 through a(5) = 16 trees:
1 2 3 4 5
(11) (111) (22) (11111)
(1(11)) (1111) ((11)3)
(2(11)) (1(22))
(1(111)) (2(111))
(11(11)) (1(1111))
((11)(11)) (11(111))
(1(1(11))) (111(11))
(1(2(11)))
(2(1(11)))
(1(1(111)))
(1(11)(11))
(1(11(11)))
(11(1(11)))
(1((11)(11)))
(1(1(1(11))))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]], disjointQ[DeleteCases[#, _?AtomQ]]&&SameQ@@Select[#, AtomQ]&], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n];
Table[Length[usot[n]], {n, 12}]
CROSSREFS
The non-locally disjoint version is A141268. Locally disjoint trees counted by vertices are A316473. The case where all leaves are 1's is A316697. Number of trees counted by A331678 with all atoms equal to 1. Matula-Goebel numbers of locally disjoint rooted trees are A316495. Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680. Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.
Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.
+10
12
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
COMMENTS
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.
EXAMPLE
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))))
MATHEMATICA
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}]
CROSSREFS
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.
Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.
+10
12
1, 2, 6, 26, 39, 78, 202, 303, 334, 501, 606, 794, 1002, 1191, 1313, 2171, 2382, 2462, 2626, 3693, 3939, 3998, 4342, 4486, 5161, 5997, 6513, 6729, 7162, 7386, 7878, 8914, 10322, 10743, 11994, 12178, 13026, 13371, 13458, 15483, 15866, 16003, 16867, 18267, 19286
COMMENTS
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. It is an identity tree if the branches under 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, two, and all nonprime squarefree numbers whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.
FORMULA
Intersection of A276625 (identity trees) and A331935 (semi-lone-child-avoiding).
EXAMPLE
The sequence of all semi-lone-child-avoiding rooted identity trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
6: (o(o))
26: (o(o(o)))
39: ((o)(o(o)))
78: (o(o)(o(o)))
202: (o(o(o(o))))
303: ((o)(o(o(o))))
334: (o((o)(o(o))))
501: ((o)((o)(o(o))))
606: (o(o)(o(o(o))))
794: (o(o(o)(o(o))))
MATHEMATICA
msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000], msiQ]
CROSSREFS
A subset of A276625 (MG-numbers of identity trees). Not requiring an identity tree gives A331935. The locally disjoint version is A331937. These trees are counted by A331964. Matula-Goebel numbers of identity trees are A276625. Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees are A331965. Cf. A004111, A007097, A050381, A061775, A196050, A291636, A300660, A306202, A320269, A331681, A331873, A331875, A331933, A331934, A331936.
Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.
+10
11
1, 1, 1, 2, 4, 6, 12, 19, 35, 59, 104, 179, 318, 556, 993, 1772, 3202, 5807, 10643, 19594, 36380, 67915
COMMENTS
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.
EXAMPLE
The a(1) = 1 through a(8) = 19 trees:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo)
(o(o)) (o(oo)) (o(ooo)) (o(oooo)) (o(ooooo))
(oo(o)) (oo(oo)) (oo(ooo)) (oo(oooo))
((o)(o)) (ooo(o)) (ooo(oo)) (ooo(ooo))
(o(o)(o)) (oooo(o)) (oooo(oo))
(o(o(o))) ((oo)(oo)) (ooooo(o))
(o(o(oo))) (o(o(ooo)))
(o(oo(o))) (o(oo)(oo))
(oo(o)(o)) (o(oo(oo)))
(oo(o(o))) (o(ooo(o)))
((o)(o)(o)) (oo(o(oo)))
(o((o)(o))) (oo(oo(o)))
(ooo(o)(o))
(ooo(o(o)))
(o(o)(o)(o))
(o(o(o)(o)))
(o(o(o(o))))
(oo((o)(o)))
((o)((o)(o)))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
strutsemi[n_]:=If[n==1, {{}}, If[n==2, {{{}}}, Select[Join@@Function[c, Union[Sort/@Tuples[strutsemi/@c]]]/@Rest[IntegerPartitions[n-1]], disjointQ]]];
Table[Length[strutsemi[n]], {n, 8}]
CROSSREFS
Not requiring lone-child-avoidance gives A316473. The Matula-Goebel numbers of these trees are A331873. The same trees counted by number of leaves are A331874. Not requiring local disjointness gives A331934. Lone-child-avoiding rooted trees are A001678. Cf. A000081, A050381, A316696, A316697, A331678, A331679, A331681, A331686, A331687, A331871, A331935.
One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.
+10
6
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 71, 74, 76, 77, 79, 80, 82, 85, 86, 88, 89, 93, 94, 95, 96, 101
COMMENTS
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 Matula-Goebel numbers of locally disjoint rooted semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. 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.
EXAMPLE
The sequence of all locally disjoint rooted semi-identity trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
6: (o(o))
7: ((oo))
8: (ooo)
10: (o((o)))
11: ((((o))))
12: (oo(o))
13: ((o(o)))
14: (o(oo))
15: ((o)((o)))
16: (oooo)
17: (((oo)))
19: ((ooo))
20: (oo((o)))
22: (o(((o))))
24: (ooo(o))
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
deQ[n_]:=n==1||PrimeQ[n]&&deQ[PrimePi[n]]||CoprimeQ@@primeMS[n]&&And@@deQ/@primeMS[n];
Select[Range[100], deQ]
CROSSREFS
The non-semi identity tree case is A316494. The enumeration of these trees by vertices is A331783. Semi-identity trees are counted by A306200. Matula-Goebel numbers of semi-identity trees are A306202. Locally disjoint rooted trees are counted by A316473. Matula-Goebel numbers of locally disjoint rooted trees are A316495.
a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).
+10
6
1, 2, 6, 26, 202, 2462, 43954, 1063462, 33076174, 1270908802, 58596709306, 3170266564862, 197764800466826, 14024066291995502, 1117378164606478094
COMMENTS
Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted identity trees. A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex. It is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an identity tree, the 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.
EXAMPLE
The sequence of terms together with their associated trees begins:
1: o
2: (o)
6: (o(o))
26: (o(o(o)))
202: (o(o(o(o))))
2462: (o(o(o(o(o)))))
MATHEMATICA
msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000], msiQ]
CROSSREFS
The semi-identity tree version is A331681. Not requiring an identity tree gives A331873. Not requiring local disjointness gives A331963. Not requiring lone-child-avoidance gives A316494. MG-numbers of semi-lone-child-avoiding rooted trees are A331935. Cf. A007097, A061775, A276625, A316471, A316495, A316694, A331679, A331683, A331686, A331872, A331934, A331936, A331964, A331965.
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