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
Keywords
Examples
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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..10000
Programs
-
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 *) -
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
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