A317588 -id:A317588 - cp's OEIS frontend

A317712 Number of uniform rooted trees with n nodes.

1, 1, 2, 4, 8, 15, 35, 72, 169, 388, 934, 2234, 5508, 13557, 33883, 85017, 215091, 546496, 1396524, 3582383, 9228470, 23852918, 61857180, 160871716, 419516462, 1096671326, 2873403980, 7544428973, 19847520789, 52308750878, 138095728065, 365153263313, 966978876376

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

An unlabeled rooted tree is uniform if the multiplicities of the branches directly under any given node are all equal.

			The a(5) = 8 uniform rooted trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  ((ooo))
  (o((o)))
  (o(oo))
  ((o)(o))
  (oooo)

Vaclav Kotesovec, Table of n, a(n) for n = 1..2250 (terms 1..200 from Andrew Howroyd)
Gus Wiseman, All 72 uniform rooted trees with 8 nodes.

Cf. A000081, A001190, A004111, A072774, A301700, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317717, A317718.

purt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],SameQ@@Length/@Split[#]&],{ptn,IntegerPartitions[n-1]}];
Table[Length[purt[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sumdiv(n-1, d, t[d]))); v} \\ Andrew Howroyd, Aug 28 2018

a(n) ~ c * d^n / n^(3/2), where d = 2.774067238136373782458114960391469140405537808253... and c = 0.43338208953061974806801546569720246018271214... - Vaclav Kotesovec, Sep 07 2019

Term a(21) and beyond from Andrew Howroyd, Aug 28 2018

A332277 Number of widely totally normal integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 2, 4, 4, 6, 3, 5, 7, 6, 8, 12, 9, 12, 13, 11, 12, 18, 17, 12, 32, 19, 25, 33, 30, 28, 44, 33, 43, 57, 51, 60, 83, 70, 83, 103, 96, 97, 125, 117, 134, 157, 157, 171, 226, 215, 238, 278, 302, 312, 359, 357, 396, 450, 444, 477, 580

Offset: 0

Gus Wiseman, Feb 12 2020

nonn

A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.

Also the number of widely totally normal reversed integer partitions of n.

			The a(n) partitions for n = 1, 4, 10, 11, 16, 18:
  1  211   4321        33221        443221            543321
     1111  33211       322211       4432111           4333221
           322111      332111       1111111111111111  4432221
           1111111111  11111111111                    4433211
                                                      43322211
                                                      44322111
                                                      111111111111111111

Normal partitions are A000009.
Taking multiplicities instead of run-lengths gives A317245.
Constantly recursively normal partitions are A332272.
The Heinz numbers of these partitions are A332276.
The case of all compositions (not just partitions) is A332279.
The co-strong version is A332278.
The recursive version is A332295.
The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.

recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],recnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

a(61)-a(66) from Jinyuan Wang, Jun 26 2020

A317718 Number of uniform relatively prime rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 27, 55, 125, 278, 650, 1510, 3624, 8655, 21017, 51212, 125857, 310581, 770767, 1920226

Offset: 1

Gus Wiseman, Aug 05 2018

An unlabeled rooted tree is uniform and relatively prime iff either it is a single node or a single node with a single uniform relatively prime branch, or the branches of the root have empty intersection (relatively prime) and equal multiplicities (uniform) and are themselves uniform relatively prime trees.

			The a(6) = 13 uniform relatively prime rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  (((ooo)))
  ((o((o))))
  ((o(oo)))
  ((oooo))
  (o(((o))))
  (o((oo)))
  (o(o(o)))
  (o(ooo))
  ((o)((o)))
  (ooooo)

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000081, A001190, A004111, A072774, A301700, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317712, A317717.

purt[n_]:=purt[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,And[SameQ@@Length/@Split[#],Intersection@@#=={}]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,20}]

A317717 Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 22, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 41, 42, 43, 47, 51, 53, 55, 58, 59, 62, 64, 66, 67, 70, 77, 78, 79, 82, 85, 86, 93, 94, 95, 100, 101, 102, 105, 106, 109, 110, 113, 114, 118, 119, 123, 127, 128

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform relatively prime tree number iff either n = 1 or n is a prime number whose prime index is a uniform relatively prime tree number, or n is a power of a squarefree number whose prime indices are relatively prime and are themselves uniform relatively prime tree numbers. A prime index of n is a number m such that prime(m) divides n.

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000081, A061775, A072774, A214577, A276625, A277098, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317712, A317718.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[200],rupQ]

A317589 Heinz numbers of uniformly normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 150, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

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

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317492, A317588, A317590.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],uninrmQ[primeMS[#]]&]

A317590 Heinz numbers of integer partitions that are not uniformly normal.

Original entry on oeis.org

10, 14, 15, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

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

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

			Sequence of all non-uniformly normal integer partitions begins: (31), (41), (32), (311), (42), (51), (2111), (61), (411), (52), (71), (43), (81), (62), (3111), (421), (511), (322), (91), (21111), (331).

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317493, A317588, A317589.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],!uninrmQ[primeMS[#]]&]

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A317712 Number of uniform rooted trees with n nodes.

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A332277 Number of widely totally normal integer partitions of n.

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A317718 Number of uniform relatively prime rooted trees with n nodes.

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A317717 Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.

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A317589 Heinz numbers of uniformly normal integer partitions.

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A317590 Heinz numbers of integer partitions that are not uniformly normal.

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