Digit sums, of any value?

[Chris65]

Hello,
I have not worked this out mathematically, I am simply studying primes in my spare time, but now as I look at the table of all of the Perfect numbers found it looks as though the digit sums of the digit sums add up to 10. Except for 6 and 28 this holds for the first 8 Perfect numbers:
28 the first digit sum = 2+8=10
496=4+9+6=19, 1+9=10
8128=8+1+2+8=19, 1+9=10
33550336=3+3+5+5+0+3+3+6=28, 2+8=10
Next digit sum=64, 6+4=10
Next digit sum=55, 5+5=10
Next digit sum=73, 7+3=10
Is there a rule to explain it? Could it be used to find more Perfect numbers?
/Sincerely, Chris65

[retina]

And the digit sum of 10 is 1.

BTW: Repeated digit sum is just another way of computing the remainder when divided by 9.

Or more succinctly: digit_sumⁿ(x) = x mod 9

[Nick]

Originally Posted by Chris65

Is there a rule to explain it?

Look up the link between even perfect numbers and Mersenne primes.
Then examine it modulo 9, as retina suggests.

[R. Gerbicz]

Originally Posted by retina

BTW: Repeated digit sum is just another way of computing the remainder when divided by 9.

Repeated digit sum has a name: https://en.wikipedia.org/wiki/Digital_root .

[sweety439]

n   possible values for even perfect numbers mod n
1   0
2   0
3   0 (only for 6), 1
4   0, 2 (only for 6)
5   1, 3
6   0 (only for 6), 4
7   0 (only for 28), 1, 6
8   0, 4 (only for 28), 6 (only for 6)
9   1, 6 (only for 6)
10  6, 8
11  1, 4, 6, 10
12  4, 6 (only for 6)
13  1, 2, 3, 6 (only for 6), 8
14  0 (only for 28), 6, 8
15  1, 6 (only for 6), 13
16  0, 6 (only for 6), 12 (only for 28)
17  1, 2, 3, 6 (only for 6), 11
18  6 (only for 6), 10
19  1, 2, 3, 6 (only for 6), 7, 9 (only for 28), 10, 15
20  6 (only for 6), 8, 16
21  1, 6 (only for 6), 7 (only for 28), 13
22  4, 6, 10, 12
23  1, 3, 5, 6, 9, 13, 15, 20
24  4 (only for 28), 6 (only for 6), 16
25  1, 3, 6, 11, 16, 21 (only for 496)
26  2, 6 (only for 6), 8, 14, 16
27  1, 6 (only for 6), 10
28  0 (only for 28), 6 (only for 6), 8, 20
29  1, 3, 4, 6, 7, 8, 16, 18, 26, 28
30  6 (only for 6), 16, 28
31  0 (only for 496), 1, 6, 27, 28
32  0, 6 (only for 6), 16 (only for 496), 28 (only for 28)