A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

16, 22, 26, 32, 44, 46, 52, 56, 58, 62, 70, 74, 76, 82, 86, 88, 92, 100, 106, 112, 116, 118, 122, 128, 130, 136, 140, 142, 146, 148, 152, 158, 160, 166, 170, 172, 176, 182, 184, 194, 196, 200, 202, 206, 212, 214, 218, 224, 226, 232, 236, 242, 244, 250, 254, 256, 262, 266, 268

Offset: 1

Author

Bernard Schott, Feb 19 2023

Keywords

• nonn

Comments

Similar sequence with odd integers d is A040976 \ {0}.

Terms are even numbers that are not divisible by 3 and that are not also in A206037.

These longest corresponding APs are of the form (q, q+d) with q odd primes (see examples).

This subsequence of A359408 corresponds to the second case '2 is one less than prime 3' (see A173919); the first case is linked to A040976.

A342309(d) gives the first element of the smallest such AP with 2 elements whose common difference is a(n) = d.

Examples

			d = 16 is a term because the first longest APs of primes with common difference 16 are (3, 19), (7,23), (13, 29), ... and all have 2 elements because next elements should be respectively 35, 39 and 45 that are all composite; the first such AP that starts with A342309(16) = 3 is (3, 19).
d = 22 is a term because the first longest APs of primes with common difference 22 are (7, 29), (19, 41), (31, 53), ... and all have 2 elements because next elements should be respectively 51, 63 and 75 that are all composite; the first such AP that starts with A342309(22) = 7 is (7, 29).
		

Links

• Wikipedia, Primes in arithmetic progression.
• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

Equals A359408 \ A040976.

Cf. A047235, A123556, A206037, A342309.

Programs

Maple

filter := d -> (irem(d, 2) = 0) and (irem(d, 3) <> 0) and not isprime(3+d) or isprime(3+d) and not isprime(3+2*d) : select(filter, [`$`(1 .. 270)]);
isA360735 := d -> isA047235(d) and not isA206037(d): # Peter Luschny, Mar 03 2023

Mathematica

Select[Range[2, 270, 2], Mod[#, 3] > 0 && Nand @@ PrimeQ[{# + 3, 2*# + 3}] &] (* Amiram Eldar, Mar 03 2023 *)

PARI

isok(d) = !(d%2) && (d%3) && !(isprime(d+3) && isprime(2*d+3)); \\ Michel Marcus, Mar 03 2023

Formula

If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(11) = 2 and 11 is not a term.