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

1,1

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