A359409 Integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly four elements.

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

Offset: 1

Author

Bernard Schott, Jan 23 2023

Keywords

• nonn

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'.

Examples

			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.
		

Links

• 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.

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).

Programs

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

Formula

m is a term iff A123556(m) = 4.