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%I #27 Aug 06 2024 09:37:24
%S 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,
%T 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,
%U 1,206,1,3,14,568,3,25,1,14,3,25,1,850,1,3,14,14
%N Number of planar branching factorizations of n.
%C 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
%H Daniel Mondot, <a href="/A277130/b277130.txt">Table of n, a(n) for n = 1..9999</a>
%H A. Knopfmacher, M. E. Mays, <a href="https://citeseerx.ist.psu.edu/pdf/d7ed31ad7c11cad37442838d6614f658af539ef5">A survey of factorization counting functions</a>, International Journal of Number Theory, 1(4):563-581,(2005). See page 15.
%F a(prime^n) = A118376(n). a(product of n distinct primes) = A319122(n). - _Gus Wiseman_, Sep 11 2018
%e From _Gus Wiseman_, Sep 11 2018: (Start)
%e The a(12) = 14 planar branching factorizations:
%e 12,
%e (2*6), (3*4), (4*3), (6*2), (2*2*3), (2*3*2), (3*2*2),
%e (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
%e (End)
%t ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
%t otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
%t Table[Length[otfs[n]],{n,20}] (* _Gus Wiseman_, Sep 11 2018 *)
%o (C)
%o #include <stdio.h>
%o #include <string.h>
%o #include <math.h>
%o #define MAX 10000
%o /* Number of planar branching factorizations of n. */
%o #define lu unsigned long
%o lu nbr[MAX]; /* number of branching */
%o lu a, b, d, e; /* temporary variables */
%o lu n; lu m, p; // factors of n
%o lu x; // square root of n
%o void main(unsigned argc, char *argv[])
%o {
%o memset(nbr, 0, MAX*sizeof(lu));
%o for (b=0, n=1; n<MAX; ++n)
%o {
%o d=0;
%o x=sqrt(n);
%o for (p=2; p<=x;++p)
%o {
%o if ((n%p)==0)
%o {
%o m= n/p;
%o if (m<p) break;
%o a = nbr[p] * nbr[m];
%o b += (m==p) ? a : 2*a;
%o e = nbr[p] * (nbr[m]-1) + (nbr[p]-1) * nbr[m];
%o d += (m==p) ? e : 2*e;
%o }
%o }
%o nbr[n]=b+d/2;
%o printf("%lu %lu\n", n, nbr[n]);
%o b = 1;
%o }
%o } /* _Daniel Mondot_, May 19 2017 */
%Y Cf. A277120.
%Y Cf. A000108, A001055, A074206, A118376, A281113, A319122, A319123.
%Y Cf. A277130, A281118, A281119, A292504, A292505, A295279, A295281, A319136, A319137, A319138.
%K nonn
%O 1,4
%A _Michel Marcus_, Oct 01 2016
%E Terms a(65) and beyond from _Daniel Mondot_, May 19 2017
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