[seqfan] Re: A057652 has no more terms below 10^4

[David Wilson]

Thank you for your positive review of my SOTD, but I am going to
have to make some disappointing conjectures.

First, I strongly doubt that you are going to find additional elements of
A057652.

I reason as follows. The original creators of the lucky numbers defined
them with an eye toward the Sieve of Eratosthenes for prime numbers.
Specifically, whereas the Sieve of Eratosthenes knocks out the multiples
of p at each step, the Lucky Sieve knocks out each pth element. From
a density standpoint, the effect is similar, the sieving step on each 
element
p removes 1/pth of the remaining elements. This similarity in the sieving
process led to a similarity in asymptotics: just as with the prime numbers,
there are about n / ln n lucky numbers <= n.

When we look at the prime number analogue of A057652, which is
A039669, we see a few small elements <= 105, and no further elements
up to 2^77 = 1.5*10^23. A heuristic argument based on the asymptotics
of the prime number argues that A039669 should indeed be finite. The
lucky number have the same asymptotics as the primes, so in absence of
any evidence to the contrary, we should expect A057652 to be finite as
well by the same heuristic argument. If you were to find another element of
A057652, I would grant that you are indeed very lucky.

My second disappointing conjecture is that, even though A057652 is
almost certainly finite, you will not find a proof. The literature on the 
prime
numbers is extensive, and yet progress in additive prime number theory
is disappointing: we have not even proved the empirically solid Goldbach
or the Twin Primes conjectures, much less the finity of A039669. The
lucky numbers do not share the favorable number theoretical properties
of the prime numbers, we cannot even program an simple test of whether
a number is lucky. For this reason I don't think a proof of the finity of
A057652 is forthcoming.

----- Original Message ----- 
From: "Alonso Del Arte" <alonso.delarte at gmail.com>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Monday, October 18, 2010 11:08 AM
Subject: [seqfan] Re: A057652 has no more terms below 10^4


1/743rd of infinity is still an infinity, but at least to me it feels
somehow more manageable. There is value to testing those largish small
numbers that are within the reach of our tools even if no proof suggests
itself in the process. In this particular case, I would wager that the
proof, if it can be found, employs modular arithmetic in some way.

David Wilson's write-up for the Sequence of the Day today shows an
encouraging example where number crunching refutes a very obvious and
reasonable assumption.

Al 



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