[seqfan] Re: Who else has looked into Erdős-Nicolas numbers?
franktaw at netscape.net
franktaw at netscape.net
Wed Aug 24 07:43:30 CEST 2011
Alonso,
It would have helped a lot if you had included the definition; this
isn't something everybody knows.
I'm assuming that these are the members of A064510 that are not perfect.
If so, there is at least one term there not in your list.
Franklin T. Adams-Watters
-----Original Message-----
From: Alonso Del Arte <alonso.delarte at gmail.com>
The known Erdős-Nicolas numbers are 24, 2016, 8190, 42336, 45864,
392448,
714240, 1571328, a result I verified with Mathematica in about 20
minutes.
Any more than that is too much for my system with my algorithm.
In the English translation of Koninck's fascinating book, the definition
explicitly says n is not a perfect number; this rules out m = 1 + n/2
as a
value in the sum_{d | n, d < m} d. The author (or translator) saw no
need to
explicitly remove deficient numbers from consideration.
But I thought, why not instead define the sum as k < tau(n) – 1, sum_{i
=
1}^k d_i, where d_1 = 1 and then in order up to d_tau(n) = n? This
rules out
perfect numbers without having the definition say "n is not a perfect
number." (Initially I was going to ask if an odd perfect number could be
an Erdős-Nicolas number, but once I started writing this message I
realized
the answer is: of course not).
However, I have not read the Erdős-Nicolas paper and as far as I can
tell
it's only available in French, so I'm not sure if there is a deeper
reason
for explicitly removing perfect numbers from consideration.
Al
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