OFFSET

1,4

COMMENTS

Per the formula, a(n) = 1 at prime n. As the sequence extends, additional values become more frequent than 1. These values can be characterized, for example, a(n) = 19 is seen at n corresponding to A007304, a(n) = 3 is seen at n corresponding to A006881, a(n) = 113 is seen at n corresponding to A085987. - Bill McEachen, Dec 28 2023

From Antti Karttunen, Jan 02 2024: (Start)

The value of a(n) depends only on the prime signature of n. In other words, for all i, j >= 1, it holds that A101296(i) = A101296(j) => a(i) = a(j). Moreover, it seems that the converse proposition also holds, that for all i, j >= 1, a(i) = a(j) => A101296(i) = A101296(j), i.e., for each distinct prime signature there exists a distinct value of a(n). This has been empirically checked up to the first 21001 prime signatures in A025487 (see A366884), and can be proved if one can show that the latter sequence (equally: A366377) is injective. If this conjecture holds, it would imply an unlimited number of statements like those given in the previous comment (see the formula section of A101296).

Questions: Are there any terms of the form 10k+4 or 10k+6? What is the asymptotic density of terms of the form 10k+5 (those ending with digit "5")? Compare to the data shown in A366884.

For squarefree n > 1, a(n) is never even, and apparently, never a multiple of five. See comments in A052886.

(End)

LINKS

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

H.-K. Hwang, Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques, 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 < n} a(d)*a(n/d). - Antti Karttunen, Nov 02 2016, after Daniel Mondot's C program, simplified Dec 30 2023.

For all n >= 1, a(prime^n) = A007317(n), and a(product of n distinct primes) = A052886(n). - Antti Karttunen, Dec 31 2023

EXAMPLE

In this scheme, the following factorizations of 12 are counted as distinct: 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.

The following factorizations of 30 are counted as distinct: 30, 2 x 15, 15 x 2, 3 x 10, 10 x 3, 5 x 6, 6 x 5, 2 x (3 x 5), 2 x (5 x 3), 3 x (2 x 5), 3 x (5 x 2), 5 x (2 x 3), 5 x (3 x 2), (2 x 3) x 5, (2 x 5) x 3, (3 x 2) x 5, (3 x 5) x 2, (5 x 2) x 3, (5 x 3) x 2, thus a(30) = 19. - Antti Karttunen, Jan 02 2024

MATHEMATICA

v[n_] := v[n] = If[n == 1, 0, 1 + Sum[If[d == 1 || d^2 > n, 0, If[d^2 == n, 1, 2]*v[d]*v[n/d]], {d, Divisors[n]}]]; Table[v[n], {n, 1, 100}] (* Vaclav Kotesovec, Jan 13 2024, after Antti Karttunen *)

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

KEYWORD

nonn

AUTHOR

Michel Marcus, Oct 01 2016

EXTENSIONS

More terms from Daniel Mondot, Oct 01 2016

STATUS

approved