Original Source A about 12volts , 15 amps PULSING dc current , #16 gauge wire electromagnet 13 turns with steel core, for X amount of magnetic field. My source as only 4 amps available as PWM at 12 volts pulsing circuit . How many turns required in same LENGTH space to approx , same original required magnetic field? 0-4amps would then be variable capability field. Smaller Gauge wire also figured in for max 4 amps.
If windings are doubled, only 1/2 current required for same field.

 

You want the current to be 4/15. 4A not 15A
The turns need to be 49/13. 49 turns not 13 turns.
The Amps X Turns need to be about the same.

 

There migh be a resistance problem, if the resistance of 49 turns is limiting the current. BUT that is an "IF" that we do not know.

 

If you reduce the current from 15 A to 4 A, you’ll need proportionally more turns to get the same magnetic field in the same length. Basically, the magnetic field B∼N⋅IB \sim N \cdot IB∼N⋅I, so if you drop the current by about 4×, you’d need roughly 4× the turns. Using thinner wire works too, just make sure it can handle the current and fits in your coil length.

 

Original Source A about 12volts , 15 amps PULSING dc current

With “pulsing” DC current into an inductor, the current will largely be determined by the frequency and duty cycle of the driving waveform with the inductance equation:

dI/dt = V/L.

 

What I don't think changes is THE POWER required to produce a given magnetic field strength. For any given constant source VOLTAGE, the current depends on the resistance, while the inductance ALSO places a limit on the rate of current increase at each pulse. Since we have no hint about the pulse rate there is no guess as to the actual average current, and since the inductance will be different with a different number of turns, there is another unknown.