[seqfan] Who else has looked into Erdős-Nicolas numbers?
Alonso Del Arte
alonso.delarte at gmail.com
Wed Aug 24 06:44:57 CEST 2011
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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