A numerical correlation between the numbers of divisors; the value of the muptiplication of the divisors, and the value of the number

It is known that the number of the positive divisors is odd IFF the integer is a square. But it is perhaps less known that it is true for every positive integer (except for those numbers where the number of the divisors is odd and where another rule applies) that
N=n^(d(n)/2) where
N=the value of the multipication of all divisors
n=number
d(n)=number of divisors (including the number n itself and 1)

E.g. if n=6 and d(n)=4 (since the divisors of 6: 1,2,3,6) then
N=6^(4/2)=36
In other words: the multiplication of the divisors result a number that is a power of the number 

Notice that in the case of the square numbers the equation is sqrt(n)^d(n).
E.g. if n=4 then N=sqrt(4)^2 (that is 8).

n	N		d(n) 	d(n)/2
1	1^1		1		-- 	(1 has only one divisor, so this is an exception)
2	2^1		3		--	
3	3^1		2		1
4	(2^3)			-- 
5	5^1		2		1
6	6^2		4		2
7	7^1		2		1
8	8^2		4		2
9	(3^3)	3		-- 
10	10^2	4		2
11	11^1	2		1
12	12^3	6		3
13	13^1	2		1
14	14^2	4		2
15	15^2	4		2
16	(4^5)	5		-- 
17	17^1	2		1
18	18^3	6		3
19	19^1	2		1
20	20^3	6		3
21	21^2	4		2
22	22^2	4		2
23	23^1	2		1
24	24^4	8		4
25	(5^3)	3		-- 
26	26^2	4		2
27	27^2	4		2
28	28^3	6		3
29	29^1	2		1
30	30^4	8		4
31	31^1	2		1
32	32^3	6		3
33	33^2	4		2
34	34^2	4		2
35	35^2	4		2
36	(6^9)	9		-- 
37	37^1	2		1
38	38^2	4		2
39	39^2	4		2
40	40^4	8		4
41	41^1	2		1
42	42^4	8		4
43	43^1	2		1
44	44^3	6		3
45	45^3	6		3
46	46^2	4		2
47	47^1	2		1
48	48^5	10		5
49	(7^3)	3		-- 
50	50^3	6		3
...

Numbers where d(n) becomes larger than earlier (we can call them using a neologism "ample numbers"):
notice that except for 4  and 36, every ample number is even. It is not a surprise, since the square numbers' n(d) is odd.

n n(d)
2 2
4 3
6 4
12 6
24 8
36 9
48 10
60 12
120 16
180 18
240 20
360 24
720 30
840 32
1260 36
1680 40
2520 48
5040 60
7560 64
10080 72
15120 80
20160 84
25200 90
27720 96
45360 100
50400 108
55440 120
83160 128
110880 144
166320 160
221760 168
277200 180
332640 192
498960 200
554400 216
665280 224
720720 240

Zoltan Galantai April 22, 2018 (modified: April 28, 2018)