A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

1, 2, 3, 10, 5, 36, 7, 120, 45, 100, 11, 936, 13, 196, 225, 3876, 17, 3078, 19, 4200, 441, 484, 23, 62400, 325, 676, 3654, 11368, 29, 27000, 31, 376992, 1089, 1156, 1225, 443556, 37, 1444, 1521, 459200, 41, 74088, 43, 43560, 46575, 2116, 47, 11995200, 1225

Offset: 1

Author

Paul D. Hanna, Aug 04 2009

Keywords

• nonn

Comments

Also the number of length n - 1 chains of divisors of n. - Gus Wiseman, May 07 2021

Examples

			Successive Dirichlet self-convolutions of the all 1's sequence begin:
(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
...
where the main diagonal forms this sequence.
From _Gus Wiseman_, May 07 2021: (Start)
The a(1) = 1 through a(5) = 5 chains of divisors:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)
      (2)  (3/1)  (2/1/1)  (5/1/1/1)
           (3/3)  (2/2/1)  (5/5/1/1)
                  (2/2/2)  (5/5/5/1)
                  (4/1/1)  (5/5/5/5)
                  (4/2/1)
                  (4/2/2)
                  (4/4/1)
                  (4/4/2)
                  (4/4/4)
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Enrique Pérez Herrero)

Crossrefs

Main diagonal of A077592.

Diagonal n = k + 1 of the array A334997.

The version counting all multisets of divisors (not just chains) is A343935.

A000005 counts divisors.

A001055 counts factorizations (strict: A045778, ordered: A074206).

A001221 counts distinct prime factors.

A001222 counts prime factors with multiplicity.

A067824 counts strict chains of divisors starting with n.

A122651 counts strict chains of divisors summing to n.

A146291 counts divisors of n with k prime factors (with multiplicity).

A167865 counts strict chains of divisors > 1 summing to n.

A253249 counts nonempty strict chains of divisors of n.

A251683/A334996 count strict nonempty length-k divisor chains from n to 1.

A337255 counts strict length-k chains of divisors starting with n.

A339564 counts factorizations with a selected factor.

A343662 counts strict length-k chains of divisors (row sums: A337256).

Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.

Cf. A060690.

Programs

Mathematica

Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* Enrique Pérez Herrero, Dec 25 2013 *)

PARI

{a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}

Python

from math import prod, comb
from sympy import factorint
def A163767(n): return prod(comb(n+e-1,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

Formula

a(p) = p for prime p.

a(n) = n^k when n is the product of k distinct primes (conjecture).

a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.

a(2^n) = A060690(n). - Alois P. Heinz, Jun 12 2024