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

1,2

Comments

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

Links

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

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.

Example

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)

Mathematica

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

Prog

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

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.

Sequence in context: A078730 A334628 A292239 * A343936 A336091 A347835

Adjacent sequences: A163764 A163765 A163766 * A163768 A163769 A163770

Keyword

nonn

Author

Paul D. Hanna, Aug 04 2009

Status

approved