A290822 -id:A290822 - cp's OEIS frontend

A290689 Number of transitive rooted trees with n nodes.

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 143, 229, 370, 592, 955, 1527, 2457, 3929, 6304, 10081

Offset: 1

Gus Wiseman, Oct 19 2017

A rooted tree is transitive if every proper terminal subtree is also a branch of the root. First differs from A206139 at a(13) = 143.

Regarding the notation, a rooted tree is a finite multiset of rooted trees. For example, the rooted tree (o(o)(oo)) is short for {{},{{}},{{},{}}}. Each "o" is a leaf. Each pair of parentheses corresponds to a non-leaf node (such as the root). Its contents "(...)" represent a branch. - Gus Wiseman, Nov 16 2024

			The a(7) = 8 7-node transitive rooted trees are: (o(oooo)), (oo(ooo)), (o(o)((o))), (o(o)(oo)), (ooo(oo)), (oo(o)(o)), (oooo(o)), (oooooo).

Robert P. P. McKone, The transitive rooted trees with n=1 to n=22 nodes.

The restriction to identity trees (A004111) is A279861, ranks A290760.
These trees are ranked by A290822.
The anti-transitive version is A306844, ranks A324758.
The totally transitive case is A318185 (by leaves A318187), ranks A318186.
A version for integer partitions is A324753, for subsets A324736.
The ordered version is A358453, ranks A358457, undirected A358454.
Cf. A000081, A001192, A001678, A324770, A324969, A325705, A358455, A358460.

nn=18;
rtall[n_]:=If[n===1,{{}},Module[{cas},Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])]]];
Table[Length[Select[rtall[n],Complement[Union@@#,#]==={}&]],{n,nn}]

a(20) from Robert Price, Sep 13 2018

a(21)-a(22) from Robert P. P. McKone, Dec 16 2023

A109082 Depth of rooted tree having Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 1, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 2, 4, 3, 5, 1, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 4, 2, 2, 3, 3, 3, 2, 2, 4, 2, 2, 4, 4, 3, 3, 5, 2, 1, 3, 4, 3, 3, 3, 3, 4, 2, 3, 3, 3, 2, 4, 3, 5, 3, 2, 4, 4, 2, 3, 3, 4, 4, 3, 3, 3, 3, 5, 4, 3, 2, 4, 2, 4, 3

Offset: 1

Keith Briggs, Aug 17 2005

nonn

Another term for depth is height.

Starting with n, a(n) is the number of times one must take the product of prime indices (A003963) to reach 1. - Gus Wiseman, Mar 27 2019

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2.

François Marques, Table of n, a(n) for n = 1..10000 (terms 1..5381 from Alois P. Heinz).
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

A left inverse of A007097.
Cf. A003963, A061775, A091233.
Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032.
This statistic is counted by A034781, ordered A080936.
The ordered version is A358379.
For node-height instead of edge-height we have A358552.
Cf. A000108, A055277, A056239, A342507, A358576.

with(numtheory): a := proc(n) option remember; if n = 1 then 0 elif isprime(n) then 1+a(pi(n)) else max((map (p->a(p), factorset(n)))[]) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, Sep 16 2011

a [n_] := a[n] = If[n == 1, 0, If[PrimeQ[n], 1+a[PrimePi[n]], Max[Map[a, FactorInteger[n][[All, 1]]]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 06 2014, after Emeric Deutsch *)

a(n) = my(v=factor(n)[,1],d=0); while(#v,d++; v=fold(setunion, apply(p->factor(primepi(p))[,1]~, v))); d; \\ Kevin Ryde, Sep 21 2020

from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A109082(n):
    if n == 1 : return 0
    if isprime(n): return 1+A109082(primepi(n))
    return max(A109082(p) for p in primefactors(n)) # Chai Wah Wu, Mar 19 2022

a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=t*s, then a(n) = max(a(t),a(s)). The Maple program is based on this.
a(A007097(n)) = n.
a(n) = A358552(n) - 1. - Gus Wiseman, Nov 27 2022

Edited by Emeric Deutsch, Sep 16 2011

A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 65, 66, 71, 72, 74, 75, 78, 79, 80, 82, 87, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 111, 113, 116, 117, 120, 122

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

A positive integer is an aperiodic tree number iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.

			Sequence of aperiodic rooted trees begins:
01 o
02 (o)
03 ((o))
05 (((o)))
06 (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)))))
33 ((o)(((o))))

Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.

zapQ[1]:=True;zapQ[n_]:=And[GCD@@FactorInteger[n][[All,2]]===1,And@@zapQ/@PrimePi/@FactorInteger[n][[All,1]]];
Select[Range[100],zapQ]

A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Mar 17 2019

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. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.

Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

A325755 Numbers n divisible by their prime shadow A181819(n).

Original entry on oeis.org

1, 2, 9, 12, 18, 36, 40, 60, 84, 112, 120, 125, 132, 156, 180, 204, 225, 228, 250, 252, 276, 280, 336, 348, 352, 360, 372, 396, 440, 441, 444, 450, 468, 492, 516, 520, 540, 560, 564, 600, 612, 636, 675, 680, 684, 708, 732, 760, 804, 828, 832, 840, 852, 876

Offset: 1

Gus Wiseman, May 19 2019

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions containing their multiset of multiplicities as a submultiset (counted by A325702).

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
    60: {1,1,2,3}
    84: {1,1,2,4}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   125: {3,3,3}
   132: {1,1,2,5}
   156: {1,1,2,6}
   180: {1,1,2,2,3}
   204: {1,1,2,7}
   225: {2,2,3,3}
   228: {1,1,2,8}
   250: {1,3,3,3}
   252: {1,1,2,2,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A181819, A182857, A290689, A290822, A323014, A324843, A325702, A325706, A325708, A325756.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Divisible[#,red[#]]&]

A324851 Numbers > 1 divisible by the sum of their prime indices.

Original entry on oeis.org

2, 4, 6, 12, 15, 16, 20, 30, 35, 36, 42, 48, 56, 88, 99, 112, 120, 126, 130, 135, 143, 144, 160, 162, 180, 192, 210, 216, 220, 221, 228, 231, 242, 250, 256, 270, 275, 280, 288, 297, 300, 308, 322, 330, 338, 360, 396, 400, 408, 429, 435, 440, 455, 468, 480, 493

Offset: 1

Gus Wiseman, Mar 18 2019

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. The sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.

For any k>=2, let d be a divisor of k such that d > A056239(k). Then 2^(d-A056239(k))*k is in the sequence. Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   56: {1,1,1,4}
   88: {1,1,1,5}
   99: {2,2,5}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  130: {1,3,6}
  135: {2,2,2,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003963, A036844, A056239, A112798, A120383, A290822, A306844.

Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.

filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2],t=ifactors(n)[2]) = 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

Select[Range[2,100],Divisible[#,Plus@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]]&]

isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k,1])*f[k,2]));} \\ Michel Marcus, Mar 19 2019

A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

Original entry on oeis.org

1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, 3390, 4290, 7878, 9570, 10230, 11310, 13026, 15510, 15990, 18330, 26070, 30966, 37290, 39390, 40890, 44070, 45210, 65130, 84810, 94830, 98310, 104610, 122070, 124410, 132990, 154830, 159330, 175890, 198330, 201630

Offset: 1

Gus Wiseman, Oct 19 2017

nonn

A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.

			Let o = {}. The sequence of transitive finitary sets begins:
1     o
2     {o}
6     {o,{o}}
30    {o,{o},{{o}}}
78    {o,{o},{o,{o}}}
330   {o,{o},{{o}},{{{o}}}}
390   {o,{o},{{o}},{o,{o}}}
870   {o,{o},{{o}},{o,{{o}}}}
1410  {o,{o},{{o}},{{o},{{o}}}}
3198  {o,{o},{o,{o}},{{o,{o}}}}
3390  {o,{o},{{o}},{o,{o},{{o}}}}
4290  {o,{o},{{o}},{{{o}}},{o,{o}}}
7878  {o,{o},{o,{o}},{o,{o,{o}}}}
9570  {o,{o},{{o}},{{{o}}},{o,{{o}}}}
10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}

Cf. A000081, A001192, A004111, A007097, A076146, A276625, A279861, A290689, A290822.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
transitaryQ[n_]:=Divisible[n,Times@@subprimes[n]];
nn=100000;Fold[Select,Range[nn],{finitaryQ,transitaryQ}]

A324846 Positive integers divisible by none of their prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 5673 are {2,11,18}, none of which divides 5673, so 5673 belongs to the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A324847.
Cf. A000720, A001222, A003963, A056239, A112798, A120383, A289509, A290822, A304360, A306844.
Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324848, A324849, A324850, A324852, A324853.

q:= n-> ormap(i-> irem(n, numtheory[pi](i[1]))=0, ifactors(n)[2]):
remove(q, [$1..200])[];  # Alois P. Heinz, Mar 19 2019

Select[Range[100],!Or@@Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>Divisible[#,PrimePi[p]]]&]

isok(n) = {my(f = factor(n)[,1]); for (k=1, #f, if (!(n % primepi(f[k])), return (0));); return (1);} \\ Michel Marcus, Mar 19 2019

A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 2, 1, 3, 3, 8, 7, 10, 13, 17, 19, 28, 35, 38, 51, 67, 81, 100, 128, 157, 195, 233, 285, 348, 427, 506, 613, 733, 873, 1063, 1263, 1503, 1802, 2131, 2537, 3005, 3565, 4171, 4922, 5820, 6775, 8001, 9333, 10860, 12739, 14840, 17206, 20029, 23248

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325755.

			The partition x = (4,3,1,1,1) has multiplicities (3,1,1), which are a submultiset of x, so x is counted under a(10).
The a(1) = 1 through a(11) = 7 partitions:
  (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)
       (211)         (3111)          (32111)  (5211)   (3331)    (33221)
                                     (41111)  (32211)  (6211)    (52211)
                                                       (42211)   (53111)
                                                       (43111)   (322211)
                                                       (322111)  (332111)
                                                       (421111)  (431111)
                                                       (511111)

Cf. A000041, A181819, A225486, A290689, A290822, A304360, A323014, A324736, A324748, A324753, A324843, A325254, A325755.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],submultQ[Sort[Length/@Split[#]],#]&]],{n,0,30}]

A318185 Number of totally transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 12, 17, 28, 41, 65, 96, 150, 221, 342, 506, 771, 1142, 1731, 2561, 3855, 5702, 8538, 12620, 18817, 27774, 41276, 60850, 90139

Offset: 1

Gus Wiseman, Aug 20 2018

A rooted tree is totally transitive if every branch of the root is totally transitive and every branch of a branch of the root is also a branch of the root. Unlike transitive rooted trees (A290689), every terminal subtree of a totally transitive rooted tree is itself totally transitive.

			The a(8) = 12 totally transitive rooted trees:
  (o(o)(o(o)))
  (o(o)(o)(o))
  (o(o)(ooo))
  (o(oo)(oo))
  (oo(o)(oo))
  (ooo(o)(o))
  (o(ooooo))
  (oo(oooo))
  (ooo(ooo))
  (oooo(oo))
  (ooooo(o))
  (ooooooo)
The a(9) = 17 totally transitive rooted trees:
  (o(o)(oo(o)))
  (oo(o)(o(o)))
  (o(o)(o)(oo))
  (oo(o)(o)(o))
  (o(o)(oooo))
  (o(oo)(ooo))
  (oo(o)(ooo))
  (oo(oo)(oo))
  (ooo(o)(oo))
  (oooo(o)(o))
  (o(oooooo))
  (oo(ooooo))
  (ooo(oooo))
  (oooo(ooo))
  (ooooo(oo))
  (oooooo(o))
  (oooooooo)

Cf. A000081, A001678, A004111, A279861, A290689, A290760, A290822, A318186, A318187.

totra[n_]:=totra[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[totra/@c]],Complement[Union@@#,#]=={}&],{c,IntegerPartitions[n-1]}]];
Table[Length[totra[n]],{n,20}]

cp's OEIS Frontend

A290689 Number of transitive rooted trees with n nodes.

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A109082 Depth of rooted tree having Matula-Goebel number n.

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A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

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A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

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A325755 Numbers n divisible by their prime shadow A181819(n).

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A324851 Numbers > 1 divisible by the sum of their prime indices.

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Maple

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A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

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A324846 Positive integers divisible by none of their prime indices.

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