A211776 a(n) = Product_{d | n} tau(d).

1, 2, 2, 6, 2, 16, 2, 24, 6, 16, 2, 288, 2, 16, 16, 120, 2, 288, 2, 288, 16, 16, 2, 9216, 6, 16, 24, 288, 2, 4096, 2, 720, 16, 16, 16, 46656, 2, 16, 16, 9216, 2, 4096, 2, 288, 288, 16, 2, 460800, 6, 288, 16, 288, 2, 9216, 16, 9216, 16, 16, 2, 5308416, 2

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = Product_{i=1..omega(n)} (b_i+1)!^(tau(n)/(b_i+1)), where omega(n) is the number of distinct prime factors of n, tau(n) is the number of divisors of n, and n = p_1^(b_1)*p_2^(b_2)* ... *p_{omega(n)}^(b_{omega(n)}). - Anand Rao Tadipatri, Aug 04 2020

Example

For n = 6: divisors of 6: 1, 2, 3, 6; tau(d): 1, 2, 2, 4; product _{d | n} tau(d) = 1*2*2*4 = 16, where tau = A000005.

Maple

A211776 := proc(n)
    mul( A000005(d), d=numtheory[divisors](n)) ;
end proc:
seq(A211776(n), n=1..20) ; # R. J. Mathar, Feb 13 2019

Mathematica

Table[Product[DivisorSigma[0, i], {i, Divisors[n]}], {n, 100}] (* T. D. Noe, Apr 26 2012 *)
a[1] = 1; a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, d = Times @@ (e + 1); Times @@ ((e + 1)!^(d/(e + 1)))]; Array[a, 100] (* using the Formula section,  Amiram Eldar, Aug 04 2020 *)

Prog

(PARI)
A211776(n) = { my(m=1); fordiv(n, d, m *= numdiv(d)); m };
A211776(n) = prod(d=1, n, if((n % d), 1, numdiv(d)));
\\ Antti Karttunen, May 19 2017

Crossrefs

Cf. A000005, A001221, A007425 (Sum_{d | n} tau(d)).

Sequence in context: A098555 A057562 A102628 * A325248 A036655 A319356

Adjacent sequences: A211773 A211774 A211775 * A211777 A211778 A211779

Keyword

nonn

Author

Jaroslav Krizek, Apr 20 2012

Status

approved