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A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3). 41
4, 14, 49, 86, 301, 454, 886, 1589, 1849, 3101, 3986, 6418, 9761, 13766, 13951, 19049, 22463, 26798, 31754, 48181, 51529, 57026, 75266, 85699, 93793, 100561, 111139, 128074, 137987, 196249, 199591, 203878, 263431, 295969, 298154, 302426, 426058, 448259, 452411 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This sequence should probably start with 1. Then a number k is in the sequence iff k = 1 or k = prime(x) * prime(y) with x and y already in the sequence. - Gus Wiseman, May 04 2021
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..186
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
FORMULA
The Matula tree of k is defined as follows:
matula(k):
create a node labeled k
for each prime factor m of k:
add the subtree matula(prime(m)), by an edge labeled m
return the node
EXAMPLE
From Gus Wiseman, May 04 2021: (Start)
The sequence of trees (starting with 1) begins:
1: o
4: (oo)
14: (o(oo))
49: ((oo)(oo))
86: (o(o(oo)))
301: ((oo)(o(oo)))
454: (o((oo)(oo)))
886: (o(o(o(oo))))
1589: ((oo)((oo)(oo)))
1849: ((o(oo))(o(oo)))
3101: ((oo)(o(o(oo))))
3986: (o((oo)(o(oo))))
6418: (o(o((oo)(oo))))
9761: ((o(oo))((oo)(oo)))
(End)
MATHEMATICA
nn=20000;
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
binQ[n_]:=Or[n===1, With[{m=primeMS[n]}, And[Length[m]===2, And@@binQ/@m]]];
Select[Range[2, nn], binQ] (* Gus Wiseman, Aug 28 2017 *)
PROG
(PARI) i(n)=n==2 || is(primepi(n))
is(n)=if(n<14, return(n==4)); my(f=factor(n), t=#f[, 1]); if(t>1, t==2 && f[1, 2]==1 && f[2, 2]==1 && i(f[1, 1]) && i(f[2, 1]), f[1, 2]==2 && i(f[1, 1])) \\ Charles R Greathouse IV, Mar 29 2013
(PARI) list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, if(i(p)&&i(q), listput(v, t*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Mar 29 2013
CROSSREFS
Cf. A245824 (by number of leaves).
These trees are counted by 2*A001190 - 1.
The semi-binary version is A292050 (counted by A001190).
The semi-identity case is A339193 (counted by A063895).
A000081 counts unlabeled rooted trees with n nodes.
A007097 ranks rooted chains.
A276625 ranks identity trees, counted by A004111.
A306202 ranks semi-identity trees, counted by A306200.
A306203 ranks balanced semi-identity trees, counted by A306201.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
Sequence in context: A220819 A047138 A363468 * A245824 A356121 A345326
KEYWORD
nonn
AUTHOR
Keith Briggs, Nov 02 2005
EXTENSIONS
Definition corrected by Charles R Greathouse IV, Mar 29 2013
a(27)-a(39) from Charles R Greathouse IV, Mar 29 2013
STATUS
approved

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Last modified April 19 02:12 EDT 2024. Contains 371782 sequences. (Running on oeis4.)