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A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

Sequence in context: A064946 A078730 A292239 * A128531 A123167 A141670

Adjacent sequences:  A163764 A163765 A163766 * A163768 A163769 A163770

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 04 2009

STATUS

approved

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Last modified March 23 05:36 EDT 2019. Contains 321422 sequences. (Running on oeis4.)