A301700 -id:A301700 - cp's OEIS frontend

A317785 Number of locally connected rooted trees with n nodes.

1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 42, 55, 67, 91, 109, 144, 177, 228, 281, 366, 448, 579, 720, 916, 1142

Offset: 1

Gus Wiseman, Aug 06 2018

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

			The a(11) = 12 locally connected rooted 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)(o)))
  (((o)(o))((o)(o)))
  ((o)(o)(o)(o)(o))

Gus Wiseman, All 42 locally connected rooted trees with 16 nodes.

Cf. A000081, A276625, A286518, A286520, A301700, A304714, A316473, A316475, A317077, A317078, A317708, A317787.

multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Length[csm[#]]==1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

A317852 Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.

Original entry on oeis.org

1, 1, 1, 3, 8, 26, 76, 247, 783, 2565, 8447, 28256, 95168, 323720, 1108415, 3821144, 13246307, 46158480, 161574043, 567925140, 2003653016, 7092953340, 25186731980, 89690452750, 320221033370, 1146028762599, 4110596336036, 14774346783745, 53203889807764, 191934931634880

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

Also the number of plane trees with n nodes where the sequence of branches directly under any given node has relatively prime run-lengths.

			The a(5) = 8 locally aperiodic plane trees:
  ((((o)))),
  (((o)o)), ((o(o))), (((o))o), (o((o))),
  ((o)oo), (o(o)o), (oo(o)).
The a(6) = 26 locally aperiodic plane trees:
  (((((o)))))  ((((o)o)))  (((o)oo))  ((o)ooo)
               (((o(o))))  ((o(o)o))  (o(o)oo)
               ((((o))o))  ((oo(o)))  (oo(o)o)
               ((o((o))))  (((o)o)o)  (ooo(o))
               ((((o)))o)  ((o(o))o)
               (o(((o))))  (o((o)o))
               (((o))(o))  (o(o(o)))
               ((o)((o)))  (((o))oo)
                           (o((o))o)
                           (oo((o)))
                           ((o)(o)o)
                           ((o)o(o))
                           (o(o)(o))

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

Cf. A000108, A000837, A007853, A032171, A032200, A254040, A301700, A303386, A303431, A304173, A304175, A317708, A317852.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
aperplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[aperplane/@c],aperQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[aperplane[n]],{n,10}]

Tfm(p, n)={sum(d=1, n, moebius(d)*(subst(1/(1+O(x*x^(n\d))-p), x, x^d)-1))}
seq(n)={my(p=O(1)); for(i=1, n, p=1+Tfm(x*p, i)); Vec(p)} \\ Andrew Howroyd, Feb 08 2020

a(16)-a(17) from Robert Price, Sep 15 2018

Terms a(18) and beyond from Andrew Howroyd, Feb 08 2020

A303552 Number of periodic multisets of compositions of total weight n.

Original entry on oeis.org

0, 1, 1, 3, 1, 9, 1, 18, 7, 44, 1, 119, 1, 246, 48, 585, 1, 1470, 1, 3248, 250, 7535, 1, 18114, 42, 40593, 1373, 93726, 1, 218665, 1, 493735, 7539, 1127981, 285, 2587962, 1, 5841445, 40597, 13244166, 1, 30047413, 1, 67604050, 216745, 152258273, 1, 342747130

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is periodic if its multiplicities have a common divisor greater than 1.

			The a(6) = 9 periodic multisets of compositions are:
{1,1,1,1,1,1},
{1,1,2,2}, {1,1,11,11},
{2,2,2}, {11,11,11},
{3,3}, {21,21}, {12,12}, {111,111}.

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303547, A303551.

nn=60;
ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
Table[SeriesCoefficient[ser,{x,0,n}]-Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]

A304649 Number of divisors d|n such that neither d nor n/d is a perfect power greater than 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 0, 1, 4, 2, 4, 2, 4, 4, 0, 2, 4, 2, 4, 4, 4, 2, 4, 1, 4, 0, 4, 2, 8, 2, 0, 4, 4, 4, 5, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 1, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 10, 2, 4, 4, 0, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 4, 0, 4, 2, 10, 4, 4, 4, 4

Offset: 1

Gus Wiseman, May 15 2018

nonn

			The a(36) = 5 ways to write 36 as a product of two numbers that are not perfect powers greater than 1 are 2*18, 3*12, 6*6, 12*3, 18*2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001055, A007427, A007916, A034444, A045778, A162247, A183096, A281116, A301700, A303386, A303707, A304650.

nn=1000;
sradQ[n_]:=GCD@@FactorInteger[n][[All,2]]===1;
Table[Length@Select[Divisors[n],sradQ[n/#]&&sradQ[#]&],{n,nn}]

a(n) = sumdiv(n, d, !ispower(d) && !ispower(n/d)); \\ Michel Marcus, May 17 2018

A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 3, 9, 12, 27, 42, 91, 151, 312, 550, 1099, 2026, 3999, 7527, 14804, 28336, 55641, 107737, 211851, 413508, 814971, 1600512, 3162761, 6241234

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches, and aperiodic if the multiplicities in the multiset of branches directly under any given node are relatively prime, and locally non-intersecting if the branches directly under any given node with more than one branch have empty intersection.

			The a(8) = 9 rooted trees:
  (o(o(o(o))))
  (o(o(o)(o)))
  (o(ooo(o)))
  (oo(oo(o)))
  (o(o)(o(o)))
  (ooo(o(o)))
  (o(o)(o)(o))
  (ooo(o)(o))
  (ooooo(o))

Cf. A000081, A000837, A007562, A289509, A301700, A303431, A316470, A316473, A316475, A316495, A319270.

btrut[n_]:=btrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[btrut/@c]]]/@IntegerPartitions[n-1],And[Intersection@@#=={},GCD@@Length/@Split[#]==1]&]];
Table[Length[btrut[n]],{n,30}]

A320805 Number of non-isomorphic multiset partitions of weight n in which each part, as well as the multiset union of the parts, is an aperiodic multiset.

Original entry on oeis.org

1, 1, 2, 6, 16, 55, 139, 516, 1500, 5269, 17017

Offset: 0

Gus Wiseman, Nov 07 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the positive entries in each row are relatively prime and (2) the column sums are relatively prime.
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 16 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,3,3}}
                    {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A303710, A320800-A320810, A321283.

A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 11, 8, 1, 10, 21, 27, 16, 1, 12, 38, 61, 63, 32, 1, 18, 57, 120, 162, 143, 64, 1, 22, 87, 205, 347, 409, 319, 128, 1, 28, 122, 333, 651, 950, 1000, 703, 256, 1, 32, 164, 506, 1132, 1926, 2504, 2391, 1535, 512, 1, 42, 217, 734, 1840

Offset: 1

Gus Wiseman, May 15 2018

A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.

			Triangle begins:
1
1    2
1    4    4
1    6   11    8
1   10   21   27   16
1   12   38   61   63   32
1   18   57  120  162  143   64
1   22   87  205  347  409  319  128
The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A303976.

Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Max],{n,10}]

T(n,k) = sum(j=1, n, sumdiv(j, d, sum(i=max(1, j+k-n), d, moebius(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1)))) \\ Andrew Howroyd, Jan 20 2023

T(n,k) = Sum_{j=1..n} Sum_{d|j} Sum_{i=max(1, j+k-n)..d} mu(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1). - Andrew Howroyd, Jan 20 2023

A304650 Number of ways to write n as a product of two positive integers, neither of which is a perfect power.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 0, 2, 0, 6, 0, 0, 2, 2, 2, 5, 0, 2, 2, 2, 0, 6, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 8, 0, 2, 2, 0, 2, 6, 0, 2, 2, 6, 0, 4, 0, 2, 2, 2, 2, 6, 0, 2, 0, 2, 0, 8, 2, 2, 2, 2, 0, 8, 2, 2, 2, 2, 2, 2, 0, 2

Offset: 1

Gus Wiseman, May 15 2018

nonn

			The a(60) = 8 ways to write 60 as a product of two numbers, neither of which is a perfect power, are 2*30, 3*20, 5*12, 6*10, 10*6, 12*5, 20*3, 30*2.

Cf. A000005, A001055, A007427, A007916, A034444, A045778, A162247, A183096, A281116, A301700, A303386, A303707, A304649.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
Table[Length[Select[Divisors[n],radQ[#]&&radQ[n/#]&]],{n,100}]

ispow(n) = (n==1) || ispower(n);
a(n) = sumdiv(n, d, !ispow(d) && !ispow(n/d)); \\ Michel Marcus, May 17 2018

A319270 Numbers that are 1 or whose prime indices are relatively prime and belong to the sequence, and whose prime multiplicities are also relatively prime.

Original entry on oeis.org

1, 2, 6, 12, 18, 24, 26, 48, 52, 54, 72, 74, 78, 96, 104, 108, 122, 148, 156, 162, 178, 192, 202, 208, 222, 234, 244, 288, 296, 312, 338, 356, 366, 384, 404, 416, 432, 444, 446, 468, 478, 486, 488, 502, 534, 592, 606, 624, 648, 666, 702, 712, 718, 732, 746

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

Also Matula-Goebel numbers of series-reduced locally non-intersecting aperiodic rooted trees.

			The sequence of Matula-Goebel trees of elements of this sequence begins:
   1: o
   2: (o)
   6: (o(o))
  12: (oo(o))
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  48: (oooo(o))
  52: (oo(o(o)))
  54: (o(o)(o)(o))
  72: (ooo(o)(o))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  96: (ooooo(o))

Cf. A000081, A007097, A061775, A276625, A298748, A301700, A303431, A316470, A319271.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ain[n_]:=Or[n==1,And[GCD@@primeMS[n]==1,GCD@@Length/@Split[primeMS[n]]==1,And@@ain/@primeMS[n]]];
Select[Range[100],ain]

A320807 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic.

Original entry on oeis.org

1, 1, 3, 6, 17, 41, 122, 345, 1077, 3385, 11214

Offset: 0

Gus Wiseman, Nov 07 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of entries equal to n and no zero rows or columns, in which each row and each column has relatively prime nonzero entries.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{2}}  {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{1},{1}}  {{1,2},{3,4}}
                    {{1},{2},{2}}  {{1,3},{2,3}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810.

cp's OEIS Frontend

A317785 Number of locally connected rooted trees with n nodes.

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A317852 Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.

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A303552 Number of periodic multisets of compositions of total weight n.

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A304649 Number of divisors d|n such that neither d nor n/d is a perfect power greater than 1.

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A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

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A320805 Number of non-isomorphic multiset partitions of weight n in which each part, as well as the multiset union of the parts, is an aperiodic multiset.

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A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

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A304650 Number of ways to write n as a product of two positive integers, neither of which is a perfect power.

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A319270 Numbers that are 1 or whose prime indices are relatively prime and belong to the sequence, and whose prime multiplicities are also relatively prime.