This may be a soft question, but it's just something I thought of one night before sleeping. It's not my field at all, so I am just asking out of curiosity. Has anyone studied the number which is the sum over primes $\sum{ 2^{p}}$? Its binary expansion (clearly) has a 1 in each prime^th "decimal place", and a zero everywhere else, so, it should be important in number theory I would guess.

5$\begingroup$ No, unlikely to be of interest in number theory. $\endgroup$ – Gerald Edgar May 11 '10 at 18:20

11$\begingroup$ Why not use $\sum 10^{p}$ instead, so you can remove the quotes around decimal place? Anyway, numbers of this form are nothing else than a curiosity but of little use. Even though they encode information about all primes, you need to input all the same information in the definition of the number. $\endgroup$ – Álvaro LozanoRobledo May 11 '10 at 18:20

$\begingroup$ @Alvaro: That's fair enough. I used 2 because what made me think of this was an old homework problem from a number theory course I took in Russia: Write a "formula" for the n^th prime. You can use a similar "trick" as above to do so. $\endgroup$ – David Carchedi May 11 '10 at 18:39

3$\begingroup$ Compare with the product formula for the Riemann zeta function: tinyurl.com/29bythb With the zeta function you elegantly get prime numbers combining together in all possible ways to form all natural numbers raised to the power of $s$. The whole thing is very natural. Your sum doesn't really have such properties. If you start trying to form powers of it (say) you get all kinds of "cross" terms that make it hard to assign meaning to the expansion. $\endgroup$ – Dan Piponi May 11 '10 at 19:55

1$\begingroup$ there are numbers that also encode this thread; check them: they also include a lot of variations that we are ashamed to try here ;) $\endgroup$ – Pietro Majer Oct 28 '12 at 21:16
Here is Hardy & Wright's answer from "An Introduction to the Theory of Numbers", (5th ed, p344), where they discuss a similar number:
"Although ... gives a 'formula' for the nth prime, it is not a very useful one. To calculate $p_n$ from this formula, it is necessary to know the value of $a$ correct to $2^n$ decimal places; and to do this, it is necessary to know the values of $p_1$, $p_2$, ..., $p_n$ ... There are a number of similar formulae which suffer from the same defect ... Any one of these formulae (or any similar one) would attain a different status if the exact value of the number $a$ which occurs in it could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."
You might take a look at the paper by Forenc Adorjan, "Binary Mappings of monotonic sequences and the Aronson function". It specifically discusses the number you describe.
See http://oeis.org/A051006 and http://mathworld.wolfram.com/PrimeConstant.html which cover this particular sequence.