A316503 -id:A316503 - cp's OEIS frontend

A317709 Aperiodic relatively prime tree numbers. Matula-Goebel numbers of aperiodic relatively prime trees.

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 66, 71, 72, 74, 75, 78, 79, 80, 82, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 113, 116, 120, 122, 123, 124, 127, 130

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is in the sequence iff either n = 1 or n is a prime number whose prime index already belongs to the sequence or n is not a perfect power and its prime indices are relatively prime numbers already belonging to the sequence. A prime index of n is a number m such that prime(m) divides n.

			The sequence of aperiodic relatively prime tree numbers together with their Matula-Goebel trees begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Cf. A000081, A061775, A111299, A214577, A276625, A277098, A301700, A303431, A316503.

Cf. A317705, A317707, A317708, A317710, A317711, A317712.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[GCD@@FactorInteger[n][[All,2]]==1,GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],rupQ]

A316500 Number of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 22, 46, 96, 205, 442, 976, 2146, 4789, 10719, 24202, 54841, 124967, 285724, 656011, 1510929, 3491151, 8088692, 18790084

Offset: 1

Gus Wiseman, Jul 05 2018

			The a(7) = 11 rooted identity trees:
  ((((((o))))))
  ((((o(o)))))
  (((o((o)))))
  ((o(((o)))))
  ((o(o(o))))
  (((o)((o))))
  (o((((o)))))
  (o((o(o))))
  (o(o((o))))
  ((o)(((o))))
  (o(o)((o)))

Cf. A000081, A004111, A007562, A078374, A289509, A316469, A316471, A316474, A316501, A316503.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Or[Length[#]==1,Intersection@@#=={}]&]];
Table[Length[strut[n]],{n,20}]

A316771 Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 4, 0, 1, 0, 2, 1, 4, 1, 0, 1, 1, 1, 6, 1, 1, 1, 4, 1, 4, 1, 2, 2, 1, 1, 8, 0, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 12, 1, 1, 2, 0, 1, 4, 1, 2, 1, 4, 1, 17, 1, 1, 2, 2, 1, 4, 1, 8, 0, 1, 1, 12, 1, 1

Offset: 1

Gus Wiseman, Jul 12 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty.

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

			The a(36) = 6 trees:
  (1(2(12)))
  (2(1(12)))
  (1(122))
  (2(112))
  (12(12))
  (1122)

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316503, A316651, A316652, 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_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>Intersection@@q=={},{0,Infinity}]&]],{n,100}]

A316772 Number of series-reduced locally nonintersecting rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 75, 202, 565, 1602, 4617, 13472, 39781, 118604, 356605, 1080178, 3293109, 10097356, 31118507, 96341035

Offset: 1

Gus Wiseman, Jul 12 2018

A rooted tree is series-reduced if all non-leaf nodes have at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty.

			The a(6) = 27 trees:
6,
(15),
(24),
(1(14)), (114),
(1(23)), (2(13)), (3(12)), (123),
(1(1(13))), (1(113)), (11(13)), (1113),
(1(2(12))), (1(122)), (2(1(12))), (2(112)), (12(12)), (1122),
(1(1(1(12)))), (1(1(112))), (1(11(12))), (1(1112)), (11(1(12))), (11(112)), (111(12)), (11112).

Cf. A000081, A000669, A001678, A141268, A292504, A316500, A316501, A316503, A316651, A316652, A316655.

nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],Intersection@@#=={}&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(17)-a(20) from Robert Price, Sep 14 2018

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A317709 Aperiodic relatively prime tree numbers. Matula-Goebel numbers of aperiodic relatively prime trees.

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A316500 Number of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

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A316771 Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number n.

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A316772 Number of series-reduced locally nonintersecting rooted trees whose leaves form an integer partition of n.

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Mathematica

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