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A277130 Number of planar branching factorizations of n. 6
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

Daniel Mondot, Table of n, a(n) for n = 1..9999

A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 15.

FORMULA

a(prime^n) = A118376(n). a(product of n distinct primes) = A319122(n). - Gus Wiseman, Sep 11 2018

EXAMPLE

From Gus Wiseman, Sep 11 2018: (Start)

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;

  }

} /* Daniel Mondot, May 19 2017 */

CROSSREFS

Cf. A277120.

Cf. A000108, A001055, A074206, A118376, A281113, A319122, A319123.

Cf. A277130, A281118, A281119, A292504, A292505, A295279, A295281, A319136, A319137, A319138.

Sequence in context: A329696 A145969 A140352 * A082588 A006241 A336105

Adjacent sequences:  A277127 A277128 A277129 * A277131 A277132 A277133

KEYWORD

nonn

AUTHOR

Michel Marcus, Oct 01 2016

EXTENSIONS

Terms a(65) and beyond from Daniel Mondot, May 19 2017

STATUS

approved

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Last modified September 18 18:19 EDT 2021. Contains 347533 sequences. (Running on oeis4.)