|
|
A277130
|
|
Number of planar branching factorizations of n.
|
|
7
|
|
|
0, 1, 1, 2, 1, 3, 1, 6, 2, 3, 1, 14, 1, 3, 3, 24, 1, 14, 1, 14, 3, 3, 1, 78, 2, 3, 6, 14, 1, 25, 1, 112, 3, 3, 3, 110, 1, 3, 3, 78, 1, 25, 1, 14, 14, 3, 1, 464, 2, 14, 3, 14, 1, 78, 3, 78, 3, 3, 1, 206, 1, 3, 14, 568, 3, 25, 1, 14, 3, 25, 1, 850, 1, 3, 14, 14
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. - Gus Wiseman, Sep 11 2018
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The a(12) = 14 planar branching factorizations:
12,
(2*6), (3*4), (4*3), (6*2), (2*2*3), (2*3*2), (3*2*2),
(2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
(End)
|
|
MATHEMATICA
|
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f], {f, Select[ordfacs[n], Length[#]>1&]}], n];
Table[Length[otfs[n]], {n, 20}] (* Gus Wiseman, Sep 11 2018 *)
|
|
PROG
|
(C)
#include <stdio.h>
#include <string.h>
#include <math.h>
#define MAX 10000
/* Number of planar branching factorizations of n. */
#define lu unsigned long
lu nbr[MAX]; /* number of branching */
lu a, b, d, e; /* temporary variables */
lu n; lu m, p; // factors of n
lu x; // square root of n
void main(unsigned argc, char *argv[])
{
memset(nbr, 0, MAX*sizeof(lu));
for (b=0, n=1; n<MAX; ++n)
{
d=0;
x=sqrt(n);
for (p=2; p<=x; ++p)
{
if ((n%p)==0)
{
m= n/p;
if (m<p) break;
a = nbr[p] * nbr[m];
b += (m==p) ? a : 2*a;
e = nbr[p] * (nbr[m]-1) + (nbr[p]-1) * nbr[m];
d += (m==p) ? e : 2*e;
}
}
nbr[n]=b+d/2;
printf("%lu %lu\n", n, nbr[n]);
b = 1;
}
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|