Perfect Abs - message from Tony Noe

[N. J. A. Sloane]

Tony Noe asked me to post this to the seqfan list.  NJAS

>From tonynoe at easystreet.com  Sun Jan 16 12:37:46 2005
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>From: "tonynoe at easystreet.com" <tonynoe at easystreet.com>
>To: njas at research.att.com
>Subject: RE: Perfect Abs
>Date: Sun, 16 Jan 2005 12:37:14 -0500
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>>On January 13, Ed Pegg Jr. submitted A101367 and A101366 (both of which 
>>I extended today). On the same day, T.D. Noe submitted A102531 and 
>>A102532, which appear (on the surface) to be be Ed's two sequences with 
>>the added inclusion of the known real perfects {6, 28, 496,...}.
>
>>Unfortunately, they use different Mathematica functions to determine 
>>the sequences and while I don't understand why the Mathematica 
>>functions should provide differing results in their application to 
>>complex numbers, it appears they do:
>
>
>
>I think it is especially confusing because the comments section of my
>sequence mentions the sum of divisors.  I have submitted new comments that
>fix this.
>
>My sequence uses a number-theoretic extension of the sigma function that is
>due to Spira.  A nonzero Gaussian integer has a unique factorization as u
>q1^e1 q2^e2..qn^en, where u is a unit (1,-1,i,-i), the qk are Gaussian
>primes in the first quadrant, and the ek are positive integers.  Then Spira
>defines the sum of divisors as prod_{k=1..n) (qk^(ek+1)-1)/(qk-1).  I
>consider this the natural number-theoretic extension.  Mathematica's
>DivisorSigma function uses this formula.
>
>Ed's sequence, which uses Mathematica's Divisors function, sums the
>first-quadrant divisors, which does not enjoy the nice multiplicative
>properties of Spira's sigma function.
>
>I'm out of town. If you want to pass this message on to seqfan, go ahead.
>
>Best regards,
>
>Tony
>
>
>
>
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