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.
a(2^n) = A060690(n). - Alois P. Heinz, Jun 12 2024
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))))}
(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
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.
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.
A337255 counts strict length-k chains of divisors starting with n.
A339564 counts factorizations with a selected factor.
Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.
Cf. A060690.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 04 2009
STATUS
approved