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A359410 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 6 elements.

%I A359410 #24 Feb 12 2023 05:38:06
%S A359410 30,60,90,120,180,240,270,300,330,360,390,450,480,510,540,570,600,660,
%T A359410 690,720,750,780,810,870,900,930,960,990,1020,1080,1110,1140,1170,
%U A359410 1200,1230,1290,1320,1350,1380,1410,1440,1500,1530,1560,1590,1620,1650,1710,1740
%N A359410 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 6 elements.
%C A359410 The 6 elements are not necessarily consecutive primes.
%C A359410 A342309(d) gives the first element of the smallest AP with 6 elements whose common difference is a(n) = d.
%C A359410 All the terms are positive multiples of 30 (A249674) but are not multiples of 7 and also must not belong to A206041; indeed, terms d' in A206041 correspond to the longest possible APs of primes that have exactly 7 elements with this common difference d'.
%H A359410 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 A359410 Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.
%H A359410 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%F A359410 m is a term iff A123556(m) = 6.
%e A359410 d = 30 is a term because the longest possible APs of primes with common difference d = 30 all have 6 elements; the first such APs start with 7, 107, 359, .... The smallest one is (7, 37, 67, 97, 127, 157); then 187 = 11*17.
%e A359410 d = 60 is another term because the longest possible APs of primes with common difference d = 60 all have 6 elements; the first such APs start with 11, 53, 641, .... The smallest one is (11, 71, 131, 191, 251, 311); then 371 = 7*53.
%e A359410 d = 150 is not a term because the longest possible AP of primes with common difference d = 150 is (7, 157, 307, 457, 607, 757, 907) which has 7 elements; this last one is unique.
%p A359410 filter := d -> (irem(d, 30) = 0) and (irem(d, 7) <> 0) and not (isprime(7+d) and isprime(7+2*d) and isprime(7+3*d) and isprime(7+4*d) and isprime(7+5*d) and isprime(7+6*d)): select(filter, [$(1 .. 1740)]);
%Y A359410 Cf. A123556, A342309.
%Y A359410 Subsequence of A249674.
%Y A359410 Longest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), this sequence (k=6), A206041 (k=7), no sequence for (k=8) and (k=9), A360146 (k=10), A206045 (k=11).
%K A359410 nonn
%O A359410 1,1
%A A359410 _Bernard Schott_, Jan 29 2023