[seqfan] Re: Who understands Granville numbers?

[William Marshall]

> From: Alonso Del Arte<alonso.delarte at gmail.com>
> Subject: [seqfan] Re: Who understands Granville numbers?
>
> I did read that comment, and I looked at the Maple program, and I thought
> about how to rewrite it for Mathematica. But I am still confused, still
> don't understand what the point of doing all those sums is. What does it
> mean for a number to be in S besides having gone through some convoluted
> test? What was it that motivated Andrew Granville to devise this complicated
> test in the first place? Was he aiming to gain some insight towards finding
> an odd perfect number?
>
> How would you go about seeing if 30 is S-perfect? If none of its divisors
> are S-perfect themselves, can we forget about 30 and move on to 31? But 28
> is not divisible by any S-perfect numbers (6 or 24). And what about prime
> numbers? What is the easiest way to explain their not being in this
> sequence? Or with a number like 2989441, should we first look at its
> smallest prime divisor or at its second largest divisor?

1 is in the set S.

For n>1, n is in the set S iff the sum of those proper divisors of n 
which are in the set S is <= n.

If the sum = n then n is S-perfect.

In general:

If n is a deficient number then n is in the set S.
If n is a perfect number then n is in the set S.

The smallest abundant number (12) is obviously NOT in the set S.

Larger abundant numbers may or may not be in the set S. You need to check.

Example:

The proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. All of the divisors 
except 12 are in the set S, so ignore 12. The sum is then 1+2+3+4+6+8 = 
24, so 24 is in the set S (and 24 is also S-perfect since the sum came 
out as 24).

My own calculation of the sequence of S-perfects gives:

6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, 15872, 
24576, 98304, 114688, 393216, 507904, 917504, 1040384, 1572864, 5540590, 
6291456, 7340032, 9078520, 16252928, 22528935, 25165824, 33550336, 
56918394, 58720256, 100663296, 133169152, ...

Note that a(28) = 22528935 is the first odd term in the sequence.