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 A359409 Integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly four elements. 9
 18, 24, 36, 54, 66, 72, 78, 84, 102, 108, 114, 132, 138, 144, 156, 162, 168, 174, 186, 192, 198, 204, 216, 222, 228, 234, 246, 258, 264, 276, 282, 288, 294, 306, 312, 318, 324, 336, 342, 348, 354, 366, 372, 378, 384, 396, 402, 408, 414, 432, 438, 444, 456, 462, 468, 486 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS These 4 elements are not necessarily consecutive primes. A342309(d) gives the first element of the smallest AP with 4 elements whose common difference is a(n) = d. All the terms are multiples of 6 (A008588) but are not multiples of 5 and also must not belong to A206039; indeed, terms d' in A206039 correspond to the largest possible arithmetic progression (AP) of primes that have exactly five elements with this common difference d'. LINKS Table of n, a(n) for n=1..56. Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French). Wikipedia, Primes in arithmetic progression. Index entries for sequences related to primes in arithmetic progressions. FORMULA m is a term iff A123556(m) = 4. EXAMPLE d = 18 is a term because the largest possible APs of primes with common difference d = 18 have all 4 elements; the first such APs start with 5, 43, 53, ... The smallest one is (5, 23, 41, 59) then 77 is composite. d = 24 is another term because the largest possible APs of primes with common difference d = 24 have all 4 elements; the first such APs start with 59, 79, 349, ... The smallest one is (59, 83, 107, 131) then 155 is composite. PROG (PARI) isok(d) = (d%5) && !(d%6) && !(isprime(5+d) && isprime(5+2*d) && isprime(5+3*d) && isprime(5+4*d)); \\ Michel Marcus, Jan 23 2023 CROSSREFS Cf. A123556, A342309. Subsequence of A008588. Largest AP of prime numbers with k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), this sequence (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7). Sequence in context: A092536 A214620 A360768 * A362432 A341295 A364998 Adjacent sequences: A359406 A359407 A359408 * A359410 A359411 A359412 KEYWORD nonn AUTHOR Bernard Schott, Jan 23 2023 STATUS approved

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Last modified May 18 15:59 EDT 2024. Contains 372664 sequences. (Running on oeis4.)