#1106: F(n) > n certified through 10,100

The target (erdosproblems.com/1106, OPEN). Let p(n) be the partition function and let F(n) count the distinct prime divisors of

p(1)p(2)···p(n).

The first question on the page, whether F(n) tends to infinity, is known: Schinzel and Wirsing proved the stronger bound F(n)≫log n. The remaining question asks whether F(n)>n for all sufficiently large n. We do not settle that asymptotic question, but we extend its verified finite range.

The result. Exact factorization through n=116 gives

F(115)=115,   F(116)=117,

and a contiguous prime-divisor certificate proves

F(n)≥n+1 for every 116≤n≤10,100.

Why the certificate works. By stage 116 there are exactly 117 distinct prime factors. For each successive n from 117 through 10,100, the certificate supplies a prime q_n dividing p(n) that is absent from the base set and different from every earlier q_j. At stage n these exhibit 117+(n−116)=n+1 distinct prime divisors of the cumulative product. This proves the inequality at every intermediate index, not just at the endpoint.

The verifier recomputes p(0),...,p(10,100) exactly using Euler's generalized-pentagonal recurrence. It completely factors the base window by trial division, checks every one of the 9,984 witness divisions over the integers, checks freshness, and asks PARI/GP's rigorous isprime routine to prove every witness prime. The final exhibited set has 10,101 distinct primes.

Independent audit. A hostile audit reran the whole chain with PARI/GP's built-in numbpart, rather than the author's recurrence. It also rebuilt the base partition numbers by coefficient dynamic programming and refactored them with a separate engine. Both routes reproduce the exact 115- and 117-prime base counts, all 9,984 divisibility and primality checks, and the running lower bound through 10,100.

Scope. OEIS A194259 and A194261 published data through 10,000 on the access date. The genuinely new part of this certificate is the 100-index continuation from 10,001 through 10,100, relative to those dated records and the searches preserved in the bundle. We do not compute exact values of F(n) after 116, claim absolute literature priority, or prove that F(n)>n eventually.