[seqfan] Re: Who understands Granville numbers?
Vladimir Shevelev
shevelev at bgu.ac.il
Fri Oct 29 20:13:58 CEST 2010
About an important subsequence of A118372.
It is easy to prove by induction that abundant of the form 3*2^k, k>=2, is in S iff k is odd. Using this (again by induction) we find a chain of S-perfects: 6, 24, 96,..., i.e. {3*2^(2t-1)}. Futher, my observations lead to conjecture (which, maybe, is known):
if P=2^p-1 is prime, then for every t>=1, P*2^(pt-1) is S-perfect. If to consider the first numbers of such chains, then we obtain all even perfect numbers. Thus we have a subsequence of S-perfects generated by even perfect numbers (in A000396):
6, 24, 28, 96, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, 15872,
24576, 98304, 114688, 393216, 507904, 917504, 1040384, 1572864,
6291456, 7340032, 16252928, 25165824, 33550336, 58720256,
100663296, 133169152, ...
Regards,
Vladimir
----- Original Message -----
From: William Marshall <w.r.marshall at actrix.co.nz>
Date: Thursday, October 28, 2010 22:36
Subject: [seqfan] Re: Who understands Granville numbers?
To: seqfan at list.seqfan.eu
> > 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.
>
>
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>
Shevelev Vladimir
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