[seqfan] Re: A007357 infinitary perfect number b-file
Vladimir Shevelev
shevelev at bgu.ac.il
Wed Mar 30 15:01:16 CEST 2011
Concerning the mentioned first 4 terms of A007357. Let A050376(n)=a(n), A007357(n)=b(n).
We have b(1)=6=a(1)*a(2)=2*3; b(2)=60=a(2)*a(3)*a(4)=3*4*5; b(3)=90=a(1)*a(4)*a(6)=2*5*9; b(4)=36720=a(2)*a(4)*a(6)*a(9)*a(10)=3*5*9*16*17. This is a full correspondence!
In 1981, I introduced in paper "On an analog of the Euler function", Proceeding of the North-Caucasus Center of the Academy of Sciences of the USSR (Rostov na Donu) (1981) No. 2, 13-17,
the notion of divisibility in Fermi-Dirac arithmetic (although the term "Fermi-Dirac" I introduced later, in 1996) which is coincides (as it is easy to prove) with infinitary divisibility. Namely, to every positive number n we correspond a inique set Q(n)of distinct terms of A050376 (to 1 corresponds the empty set). Then we call d a Fermi-Dirac divisor, if Q(d) is a subset of Q(n).
A number n we call Fermi-Dirac perfect, if it is sum of all
proper Fermi-Dirac divisors of n. E.g., Q(60)={3,4,5}. Its proper Fermi-Dirac divisors are 1,3,4,5,3*4,3*5,4*5 and we have 1+3+4+5+12+15+20=60.
Regards,
Vladimir
----- Original Message -----
From: Richard Mathar <mathar at strw.leidenuniv.nl>
Date: Wednesday, March 30, 2011 13:33
Subject: [seqfan] Re: A007357 infinitary perfect number b-file
To: seqfan at seqfan.eu
>
> Concerning
> http://list.seqfan.eu/pipermail/seqfan/2011-March/014690.html
> and the information currently in http://oeis.org/A007357
>
> vs> ...
> vs> I published a list of all 36 such numbers up to 2^16 by name
> "Fermi-Dirac perfect numbers".
> vs> ...
> If "such numbers" refers to A007357 it means that these first 4
> of these
> numbers were calculated, which is basically irrelevant. If "such
> numbers"refers to A050376, I do not see how there are only 36 in
> the range up to 2^16.
>
>
>
> vs> and I learned that J. M. Pedersen in "Tables of Aliquot
> Cycles." http://amicable.adsl.dk/aliquot/infper.txt. " in 2004
> calculated the first 155 iperfect numbers.
> vs>
>
> The Pedersen table in http://amicable.adsl.dk/aliquot/i1/i1.txt
> is annotated
> with "Perfect numbers, exhaustive below 5*10^12" in
> http://amicable.adsl.dk/aliquot/tables.htmThis gives rich
> opportunity to some sort of standard misinterpretation to
> some readers that this table were complete up to its last member
> listed. We
> have had this sort of trouble with OEIS submissions concerning
> Mersennenumbers and other cases, where the list of known
> candidates reaches far beyond
> the highest watermark of the dense lists.
> Here, the current entries in A007357 reach up to 3*10^16,
> further than
> the Pedersen high watermark. The answer to my original question in
> http://list.seqfan.eu/pipermail/seqfan/2011-March/014687.html is
> "No, we should not copy more numbers from the Pedersen table to
> A007357."
> Richard Mathar
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
Shevelev Vladimir
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