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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.

%I A359409 #19 Jan 29 2023 19:37:45
%S A359409 18,24,36,54,66,72,78,84,102,108,114,132,138,144,156,162,168,174,186,
%T A359409 192,198,204,216,222,228,234,246,258,264,276,282,288,294,306,312,318,
%U A359409 324,336,342,348,354,366,372,378,384,396,402,408,414,432,438,444,456,462,468,486
%N A359409 Integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly four elements.
%C A359409 These 4 elements are not necessarily consecutive primes.
%C A359409 A342309(d) gives the first element of the smallest AP with 4 elements whose common difference is a(n) = d.
%C A359409 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'.
%H A359409 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3940-a1880-np-en-pa">A1880. NP en PA</a> (prime numbers in arithmetic progression) (in French).
%H A359409 Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.
%H A359409 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%F A359409 m is a term iff A123556(m) = 4.
%e A359409 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.
%e A359409 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.
%o A359409 (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
%Y A359409 Cf. A123556, A342309.
%Y A359409 Subsequence of A008588.
%Y A359409 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).
%K A359409 nonn
%O A359409 1,1
%A A359409 _Bernard Schott_, Jan 23 2023