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A277120 Number of branching factorizations of n. 5
0, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 11, 1, 3, 3, 15, 1, 11, 1, 11, 3, 3, 1, 45, 2, 3, 5, 11, 1, 19, 1, 51, 3, 3, 3, 62, 1, 3, 3, 45, 1, 19, 1, 11, 11, 3, 1, 195, 2, 11, 3, 11, 1, 45, 3, 45, 3, 3, 1, 113, 1, 3, 11, 188, 3, 19, 1, 11, 3, 19, 1, 345, 1, 3, 11, 11, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

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

H.-K. Hwang, Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, PhD thesis, in Ecole Polytechnique, 1994. See page 198.

A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 14.

FORMULA

a(1) = 0; for n > 1, a(n) = 1 + Sum_{d|n, 1 < d <= sqrt(n)} a(d)*a(n/d)*(2 if d^2 < n, 1 otherwise). - Antti Karttunen, Nov 02 2016, after Daniel Mondot's C-program.

EXAMPLE

In this scheme, the following factorizations of 12 are counted as different: 12, 2 x 6, 2 x (2 x 3), 2 x (3 x 2), 3 x 4, 3 x (2 x 2), 4 x 3, (2 x 2) x 3, 6 x 2, (2 x 3) x 2, (3 x 2) x 2, thus a(12) = 11. - Antti Karttunen, Nov 02 2016, based on the illustration given at page 14 of Knopfmacher & Mays paper.

PROG

(C)

#include <stdio.h>

#define MAX 10000

/* Number of branching factorizations of n. */

unsigned long n, m, a, b, p, x, nbr[MAX];

int main(void)

{

  for (x=n=1; n<MAX; ++n)

  { if (x*x == n) ++x;

    for (b=0, 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;

      }

    }

    nbr[n] = b+1;

  }

  printf("1 0\n");

  for (n=2; n<MAX; ++n) printf("%lu %lu\n", n, nbr[n]);

  return(0);

} /* Daniel Mondot, Oct 01 2016 */

(PARI) A277120(n) = if(1==n, 0, 1+sumdiv(n, d, if((1==d)||(d*d)>n, 0, if((d*d)==n, 1, 2)*A277120(d)*A277120(n/d)))); \\ Antti Karttunen, Nov 02 2016, after Daniel Mondot's C-program above.

(PARI) seq(n)={my(v=vector(n)); for(n=2, n, v[n] = 1 + sumdiv(n, d, v[d]*v[n/d])); v} \\ Andrew Howroyd, Nov 17 2018

CROSSREFS

Cf. A074206.

After n=1 differs from A104725 for the next time at n=32, where a(32) = 51, while A104725(32) = 52.

Sequence in context: A085053 A296118 A296121 * A104725 A289079 A249810

Adjacent sequences:  A277117 A277118 A277119 * A277121 A277122 A277123

KEYWORD

nonn

AUTHOR

Michel Marcus, Oct 01 2016

EXTENSIONS

More terms from Daniel Mondot, Oct 01 2016

STATUS

approved

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Last modified July 28 04:08 EDT 2021. Contains 346316 sequences. (Running on oeis4.)