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To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does this imply that a good way to estimate the amount of lucky twins between $3$ and $x$ is $x \prod_{l}(1-2/l)$ where $l$ runs over the lucky numbers between $3$ and $\sqrt x$?

*EDIT:*I Think I can improve the question by stating it as follows.

$1$) If $x$ goes to infinity and $l(x)$ denotes the amount of lucky numbers between $3$ and $x$ does $\dfrac{x \prod_{l}(1-2/l)}{l(x)}=Constant$ ?

$2$) If $x$ goes to infinity, does $\dfrac{\prod_{p}(1-2/p)}{\prod_{l}(1-2/l)}=Constant$ ?

If no theoretical answer is possible are $1$) and $2$) supported numerically ?

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Note that since it is unknown whether there's even infinitely many twin primes, there's no way to be sure that your initial estimate of twin primes is any good for large $x$. Therefore it will be very hard to argue rigorously about whether the lucky-twins one works better or worse. – Henning Makholm Feb 19 '13 at 22:17
Some arguments are that lucky numbers are not building blocks by multiplication for the integers whereas primes are. Also the n'th lucky number is larger than the n'th prime for n sufficiently large. – mick Feb 19 '13 at 22:35
In the title, you ask about estimating the number of lucky twins. In the body, you ask about estimating the number of lucky numbers. Please edit for consistency. – Gerry Myerson Feb 19 '13 at 23:01
What the heck is a lucky number? – Willie Wong Feb 20 '13 at 8:52
@Willie: – joriki Feb 20 '13 at 10:08
up vote 1 down vote accepted

I think that to answer this question one would need to start from the paper by Bui and Keating on the random sieve (of which the lucky sequence is a particular realization) and then carefully examine the proofs that they cite of Mertens' theorems for that sieve. The issue is whether there is a correction factor like Mertens' constant or the twin prime constant in the product that you wrote down for the number of lucky twins, and if there is, is it calculated (up to $1+o(1)$ factors) by the same formula as for the primes.

The paper gives an asymptotic formula for random twin primes that does not have any twin prime constant, but the explanation there is not exactly about the same situation. It looks to me like some analysis of the proofs of the analogues of Mertens' theorems is needed to understand what product over random primes is the correct estimate of the number of random twin primes.

I cannot tell the answer to the question from Bui and Keating paper by itself, but would guess that up to multiplication by a constant, the product you wrote down is, for nearly all realizations of the random sieve (ie., with probability 1) asymptotic to the number of random-sieve twins, and this is the heuristic guess for what happens with the lucky numbers.

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