This item set is centered around Lux's passive, dealing additional magic damage against enemies you have recently used an ability on when you auto attack them. At level 18 the passive does 190 (+20% ap) as extra on hit magic damage. An important part is auto attacking, the faster you auto attack the more additional damage you can inflict.

Lux has mana problems and this item helps with that. When it fully upgrades, it gives 80 ap and 1000 mana, refunds 25% of mana spent.It also gives you ap equal to 3% of your max mana. With lich bane and her natural mana pool, along with a fully upgraded archangels (seraphs embrace) it is a total of 2433 mana in her pool. which gives 72.99 additional ap.

This is a common buy on lux, it gives 250 mana, 80 ap and 10% cdr. It's passive is that after an ability your next auto attack deals an additional 75% ad + 50% ap.

This is where the build gets a little weird. Nashors gives you 80 ap, 50% attack speed and 20% cdr. It has a passive that deals 15 damage (+15% ap)magic damage on hit. As I said before, the more you auto attack the more damage you can inflict using Lux's passive and now with Nashors and Lich bane's passive. That is now an 205 (+85% ap) and (+75% ad) additional auto attack damage against enemies you have recently used an ability on.

This item gives you 35 ad, 50 ap and 25% attack speed. It has two passives the first is you do an extra 15 magic damage on hit. The other is Basic attacks grant +8% Attack Speed, +3 Attack Damage, and +4 Ability Power for 5 seconds and can stack up to 6 times. I will put the second passive to the side. But basically you have an additional 15 magic damage on hit plus 6(3 ad + 4 ap), that equation I will put to the side. So with all the items so far you have 230 (+85% ap) and (75% ad) on hit effect.

This item gives you 40 magic resist, 40% attack speed, 40 bonus magic damage on hit and steals 5 magic resist per auto attack (stacking up to 5 times, 25 magic resist being stolen in total). With all the items so far you have 270 (+ 85% ap) and (75% ad) damage on hit.

So boots are usually a must, I don't recommend any specific boots. You can get Beserker's Grieves or Sorcerer's Shoes if you and your team are ahead. Boots of Swiftness or Mobility if you are getting frustrated with your movement speed, although I don't recommend Mobility since you'll be auto attacking and using your abilities quite often. If you need to be more defensive there is always Mercury's Treads and Ninja Tabi (this build lacks any armor so if you want to go a bit more defensive Ninja Tabi would be your best bet). If you want full CDR and you dont have CDR runes you can always go Ionian Boots of Lucidity.

If you want to get boots you can choose to go tanky or get extra magic damage items. Runaan's Hurricane is also an option, it gives you attack speed and an additional 15 damage on hit. As a bonus you can also hit multiple targets at once, it only deals 25% of your ad to the extra two targets but I believe your on hit effects still apply.

Aside from any sixth item, your 5 core items while they are all at their full potential give you:
AP: 362.99
AS: 0.625 (+138.1%) (1.488 attacks per sec)
Ad: 145
Mana: 2433
Health: 1821
Armor: 86.7
MR: 78.5

270 (+ 85% ap) and (75% ad)
85% of your final ap is 308.5415
75% of your final ad is 108.75

So when you auto attack a target with Lux's passive Illumination present is a total of 687.2915 extra damage.

Add in your ad to that auto attack your final damage is 832.2915 and this is ignoring whatever damage you just did to them with your abilities.

Q: 250 (+70% ap)=
504.093

W: shields for 110 (20% ap)=
182.598

E: 240 (+60% ap)=
457.794

R: 500 (+75% ap)=
772.2425

So I went ahead and got these items, with no masteries (except one in wanderer) and no runes, at level 18, and tested out my on hit effect in the practice tool in league. I forgot to take in consideration any magic penetration or some other factor that may increase or decrease damage.

So I tested the on hit effects on first spawn blue buff which has 3675 health. I used my Q and it did 625 damage. The buff is now at 3050 health. I auto attacked it once and it did 923 damage. The Buff was left with 2127 health.

So total damage with one Q and one auto is 1548 damage (not taking in account any magic resist or armor your opponent may have). I tested this multiple times and came up with the same numbers.

Everyone knows Lux is a pretty squishy character. This build does a lot of damage but at a price. Although you get more damage on your on hit effects after casting an ability, you also have to get close enough to auto attack your target. This is risky because Lux normally plays from afar and her auto attack range is shorter than her ability range. Getting close to your target may make you more vulnerable to enemy attacks.