You may reconsider your math as you only get extra mana from one Archangel's Staff -> unique passive.

So your total mana would be:

1025 + 51,25 + 1400 + 3 x 400 + 1350 = 5026,50 resulting in an extra 603 ability power from mana only; Giving you a total of 793 ability power;

Also you should put Manamune as second item after the first Archangel's Staff so it can build up its stacks more quickly.

I. Alternative

As an alternative you could also skip one Archangel's Staff for a Rabadon's Deathcap. You then would lose 400 mana, 100 hit points, (45 AP+ 187 AP= ) 232AP & 25 MP/5 and gain (140 AP + 167 AP= ) 307 AP. In total that would be 75 AP more for the loss of the other stats resulting in 868 AP total;

II. Alternative

If you would skip Manamune for a Rabadon's Deathcap in your standard build you would gain 489 AP from mana, 320 from the items and 30% extra resulting in 1052 AP which seems like the best outcome for your approach. You would lose 900 mana, 225 hp, 20 AD and 7 MP/5 for 259 AP. I would say this is the most effective outcome, even if it is everything but cheap :)

But that are just my thoughts, maybe some of this was helpful.