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%I #32 Feb 18 2023 03:37:24
%S 150,2760,3450,9150,14190,20040,21240,63600,76710,117420,122340,
%T 134250,184470,184620,189690,237060,274830,312000,337530,379410,
%U 477630,498900,514740,678750,707850,1014540,1168530,1180080,1234530,1251690,1263480,1523520,1690590
%N Values of the difference d for 7 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 6.
%C The computations were done without any assumptions on the form of d.
%C All terms are multiples of 30. - _Zak Seidov_, Jan 07 2014.
%C Equivalently, integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 7 elements (see example). These 7 elements are not necessarily consecutive primes. In fact, for each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 7, so this unique AP is (7, 7+d, 7+2d, 7+3d, 7+4d, 7+5d, 7+6d). - _Bernard Schott_, Feb 12 2023
%H Sameen Ahmed Khan, <a href="/A206041/b206041.txt">Table of n, a(n) for n = 1..10000</a>.
%H 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 Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1203.2083">Primes in Geometric-Arithmetic Progression</a>, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%F m is a term iff A123556(m) = 7. - _Bernard Schott_, Feb 12 2023
%e d = 150 then {7, 7 + 1*150, 7 + 2*150, 7 + 3*150, 7 + 4*150, 7 + 5*150, + 7 + 6*150} = {7, 157, 307, 457, 607, 757, 907} which is 7 primes in arithmetic progression.
%p filter := d -> 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 .. 1700000)]); # _Bernard Schott_, Feb 13 2023
%t a = 7; t = {}; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d, a + 6*d}] == {True, True, True, True, True, True, True}, AppendTo[t,d]], {d, 200000}]; t
%Y Cf. A040976, A206038, A206040, A206042, A206043, A206044, A206045, A123556, A342309.
%Y Longest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), this sequence (k=7), A360146 (k=10), A206045 (k=11).
%K nonn
%O 1,1
%A _Sameen Ahmed Khan_, Feb 03 2012
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