Factoring π

Well, not π exactly but integers whose binary representation are found in the binary form of π. The factoring challenges that I am aware of, (RSA Inc. and FastLizard4), will convince most that factoring is hard since the payout was substantial. It will not convince skeptics that factoring is easy for the skeptic will suggest that the challenger, who knew the factors, leaked them to the factorer in order to create a false impression at large. Here is a challenge that avoids that ploy.

For challenge n seek a composite number N,

whose bits appear as contiguous bits in binary π,
with no prime factors less than 2¹²⁸ⁿ.

I think that the density of solutions without the π constraint is about 1/log(N).

I imagine a digital contract that would pay off on presentation of such factors, along with where in π the product is found. I think that Ethereum is not up to this yet. Adopting the binary form over the decimal form may help resolve arguments as to whether N is found in π as claimed. Most would be satisfied by N appearing in the RC4 cipher stream resulting from "".