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

%I A360735 #31 Mar 03 2023 16:52:34
%S A360735 16,22,26,32,44,46,52,56,58,62,70,74,76,82,86,88,92,100,106,112,116,
%T A360735 118,122,128,130,136,140,142,146,148,152,158,160,166,170,172,176,182,
%U A360735 184,194,196,200,202,206,212,214,218,224,226,232,236,242,244,250,254,256,262,266,268
%N A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.
%C A360735 Similar sequence with odd integers d is A040976 \ {0}.
%C A360735 Terms are even numbers that are not divisible by 3 and that are not also in A206037.
%C A360735 These longest corresponding APs are of the form (q, q+d) with q odd primes (see examples).
%C A360735 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.
%C A360735 A342309(d) gives the first element of the smallest such AP with 2 elements whose common difference is a(n) = d.
%H A360735 Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.
%H A360735 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%F A360735 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.
%e A360735 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).
%e A360735 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).
%p A360735 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)]);
%p A360735 isA360735 := d -> isA047235(d) and not isA206037(d): # _Peter Luschny_, Mar 03 2023
%t A360735 Select[Range[2, 270, 2], Mod[#, 3] > 0 && Nand @@ PrimeQ[{# + 3, 2*# + 3}] &] (* _Amiram Eldar_, Mar 03 2023 *)
%o A360735 (PARI) isok(d) = !(d%2) && (d%3) && !(isprime(d+3) && isprime(2*d+3)); \\ _Michel Marcus_, Mar 03 2023
%Y A360735 Equals A359408 \ A040976.
%Y A360735 Cf. A047235, A123556, A206037, A342309.
%K A360735 nonn
%O A360735 1,1
%A A360735 _Bernard Schott_, Feb 19 2023