# tau(k)/bigomega(k)

Emeric Deutsch deutsch at duke.poly.edu
Wed Jun 29 16:23:15 CEST 2005

```Dear Seqfans,
Sequence A109421 gives the numbers k such that tau(k)/bigomega(k)
is an integer [tau(k)=number of divisors of k; bigomega(k)=number
of prime divisors of k, counted with multiplicities].

My question: does the set {tau(k)/bigomega(k): k=2,3,...} contain
all integers n>=2?

We (= Maple and I) have found that the least k such that
tau(k)/bigomega(k)=n is given by the table:
n	k
-	-
2	2
3	60
4	210
5	2160
6	1260
7	77760
8	4620
9	12600
10	18480
11	3456000
12	27720
13	4730880
14	302400
15	453600
16	120120
17	>11,000,000 if it exists
18	180180
19	>11,000,000 if it exists
20	997920
21	1108800
22	10644480
23	>11,000,000 if it exists
24	720720
25	2494800
26	>11,000,000 if it exists
27	3880800
28	2882880
29	>11,000,000 if it exists
30	5821200
31	>11,000,000 if it exists
32	3063060
33	>11,000,000 if it exists
34	>11,000,000 if it exists
35	>11,000,000 if it exists
36	7207200
37	>11,000,000 if it exists
38	>11,000,000 if it exists
39	>11,000,000 if it exists
40	10810800

I have similar questions and comparable data for A109423,A109425, and
A109427.
I'd appreciate collaborators.
Thanks.
Emeric

```