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 A309270 a(n) is the largest k such that the first k odd primes can be covered by n arithmetic progressions of primes. 2
 3, 5, 10, 13, 18, 22, 24, 27, 31, 34, 39, 41, 45, 50, 55, 62, 64, 68, 73, 79, 81, 89, 91, 96, 99, 102, 107, 110, 115, 119, 124, 128, 133, 137, 142, 145, 151, 156, 162, 166, 170, 174, 177, 182, 185, 190, 193, 199, 203, 208 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Here we allow the arithmetic progressions to contain one or more terms. The first 1000 odd primes can be covered with 221 arithmetic progressions of primes (see Links). Finding the smallest n for a given k is a set covering problem with a binary variable for each arithmetic progression and a constraint for each of the first k odd primes. - Rob Pratt, Aug 26 2019 LINKS Table of n, a(n) for n=1..50. dxdy forum, Covering of primes with arithmetic progressions of primes (in Russian) Dmitry Kamenetsky, Covering of the first 1000 odd primes Carlos Rivera, Puzzle 963: minimal quantity of prime arithmetic progressions to cover the first primes EXAMPLE 1 arithmetic progression of primes is needed to cover the first 3 odd primes: (3,5,7). So a(1) = 3. Note that we cannot cover the first 4 odd primes with 1 arithmetic progression. 2 arithmetic progressions of primes are needed to cover the first 5 odd primes: (3,7,11), (5,13). So a(2) = 5. 3 arithmetic progressions of primes are needed to cover the first 10 odd primes: (3,17,31), (5,11,17,23,29), (7,13,19). So a(3) = 10. 4 arithmetic progressions of primes are needed to cover the first 13 odd primes: (3,13,23), (5,17,29,41), (7,19,31,43), (11,37). So a(4) = 13. 5 arithmetic progressions of primes are needed to cover the first 18 odd primes: (5,11,17,23,29), (7,19,31,43), (41,47,53,59), (13,37,61), (3,67). So a(5) = 18. CROSSREFS Cf. A309095. Sequence in context: A275219 A080931 A299533 * A165718 A340528 A031878 Adjacent sequences: A309267 A309268 A309269 * A309271 A309272 A309273 KEYWORD nonn,more,hard AUTHOR Dmitry Kamenetsky, Jul 20 2019 EXTENSIONS a(27)-a(50) from Rob Pratt, Aug 26 2019 STATUS approved

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Last modified December 1 23:26 EST 2023. Contains 367503 sequences. (Running on oeis4.)