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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
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
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
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
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.)