A machinesmith can gain temporary feats using a mobious weapon. An android machinesmith can gain another temporary feat by casting on himself the spell program feat. Can this android machinesmith use these temporary feats to meet the prerequisites of other temporary or permanent feats?
A character can't use a feat if he loses a prerequisite, but he does not lose the feat itself. If, at a later time, he regains the lost prerequisite, he immediately regains full use of the feat that prerequisite enables. Feat rules
You are free to select any feat that you meet the prerequisites for, but as soon as you lose any of the prerequisites (i.e. your temporary feats no longer apply) you lose the ability to make use of that feat and any that it itself is a prerequisite for as well.
Another example of where this logic applies is to the Brawler class, as confirmed by the Pathfinder lead designer.
A brawler can use the feats granted by brawler's flurry to qualify for other feats, but can only use those other feats when using brawler's flurry (as that's the only time she actually meets those prerequisites).
From the page on Mobius Weapons
The machinesmith does not have to meet the feat's normal qualifications, but retains the feat only for a number of minutes equal to his Intelligence modifier (minimum 0), as long as they are wielding their mobius weapon.
From the page on Program Feat
Program feat gives a construct the benefits of a single feat possessed by another creature the caster is touching when the spell is cast. The construct must meet all the prerequisites of the feat.
Yes for Temporary, No for Permanent
Based on the wording of Program Feat Especially the line "must meet all prerequisites" I would say the answer is yes, as long as you meet all of the other prerequisites.
For example: Say you want to get Cleave from Program Feat. You already have 13 Str and a BAB of +1 but you do not have Power Attack. You use a charge of your Mobius Weapon to grant you Power Attack for several minutes. You now meet all of the prerequisites for Cleave and can cast Program Feat on yourself to gain Cleave. Once the charge from your Mobius Weapon wears off you lose Power Attack and Cleave. You could potentially expend another charge to regain Power Attack and retain Cleave until one or both of the effects has worn off.
As for using temporary feats as prerequisites to gain permanent feats I would say No, you still need to meet all of the prerequisites for a feat when leveling up.