Q:

How to increase the distance of effective reach of a magnetic field from an electromagnet

How can I effectively increase the distance in which some electromagnet can attract a coin (for the sake of simplicity - coin).

Now to show that I've done my homework:
According to Ampere's Law, the strength of the magnetic field is directly proportional to the number of coils. Therefore, more coils (of some sort) right?
How about pressing the coils by virtue of hydraulic press? Wouldn't that allow me to coil more wire at less space? The denser the coil is the better eh?
Also the law tells us that more current will also increase the power of the electromagnet. Well I have 4 3000 mAh 20A/3.6V li-ion LG batteries, are they sufficient? How about using OP-amps? I am not afraid of draining the batteries down quickly.

Best answer will explain why electromagnets loose strength over distance  by the cube of the radius distance from the magnet (as corrected by @Trevor), ways to increase the distance, possibly up to the length of a half hand and explain how to measure what can one achieve, given number of coils, current, coil density etc...

A:

Although air can be magnetized, its relative permeability (\$\mu_r\$) is defined as 1 — about the same as copper and aluminum — which is about as weak as you can get.  So it does not conduct magnetic flux very well compared to iron (\$\mu_r=5000\$) or annealed iron (\$\mu_r=200k\$).
In order to have a high pull strength, the gap must be very small or the number of enamel insulated turns must be very large.
Fortunately, our bodies are not very magnetic except to very high >4 Tesla levels which can be used for imaging of excited electrons spinning out of orbit.  Where I once lived they developed a portable scanner for putting a patient inside the tube in an operating room that did not have metal in it.  They can distort the images of old CRT's under a 10 story building and accelerate a metal chair across a room up highway speeds towards the cryogenically cooled coils. But that's how MRI's work.
Unfortunately Op Amps do not conduct much current needed to make an electromagnet. But beware that when you open the circuit, the inductor stores the energy which must be released at the contacts creating a huge voltage. \$V=L\frac{di}{dt}\$ and \$dt\$ can get really small with an open wire.
For flux calculations: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html#c3 
Now they are are trying to charge mobiles wirelessly and even EV's (electric vehicles) with 4kW at present, 12kW in labs and 18kW in future. In order to do this impedance matching sending and receiving coils need to be designed to minimize leakage and have a diameter at least 2x to 4x the gap.  
Less is more, as in flux coupling.

