|
|
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
|
|
|
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
|
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
|
These trees are counted by 2*A001190 - 1.
A000081 counts unlabeled rooted trees with n nodes.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|