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%I #40 Mar 09 2023 08:47:00
%S 1536160080,4911773580,25104552900,77375139660,83516678490,
%T 100070721660,150365447400,300035001630,318652145070,369822103350,
%U 377344636200,511688932650,580028072610,638663371710,701534299830,745828915650,776625236100,883476548850,925639075620,956863233690
%N Numbers d such that 11 + j*d is prime for j = 0 to 10.
%C Original name: Values of the difference d for 11 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 10.
%C The computations were done without any assumptions on the form of d. 21st term is greater than 10^12.
%C All terms are multiples of 210=2*3*5*7. - _Zak Seidov_, May 16 2015
%C Equivalently, integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 11 elements (see example). These 11 elements are not necessarily consecutive primes. In fact, here, for each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 11, so this unique AP is (11, 11+d, 11+2d, 11+3d, 11+4d, 11+5d, 11+6d, 11+7d, 11+8d, 11+9d, 11+10d). - _Bernard Schott_, Mar 08 2023
%H Zak Seidov, <a href="/A206045/b206045.txt">Table of n, a(n) for n = 1..623</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.
%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) = 11. - _Bernard Schott_, Mar 08 2023
%e d = 4911773580 then {11, 4911773591, 9823547171, 14735320751, 19647094331, 24558867911, 29470641491, 34382415071, 39294188651, 44205962231, 49117735811} which is 11 primes in arithmetic progression.
%t a = 11; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d, a + 6*d, a + 7*d, a + 8*d, a + 9*d, a + 10*d}] == {True, True, True, True, True, True, True, True, True, True, True}, Print[d]], {d, 210,10^12, 210}] (* corrected by _Zak Seidov_, May 16 2015 *)
%t Select[Range[210,10^12,210],AllTrue[Range[0,10]#+11,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 28 2016 *)
%o (PARI) is(n)=for(j=1,10, if(!isprime(j*n+11), return(0))); 1 \\ _Charles R Greathouse IV_, May 18 2015
%Y Cf. A040976, A206038, A206040, A206042, A206043, A206044.
%Y Cf. A123556, A173919.
%Y Common differences for longest possible APs of primes with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A360146 (k=10), this sequence (k=11).
%K nonn
%O 1,1
%A _Sameen Ahmed Khan_, Feb 03 2012
%E New name from _Charles R Greathouse IV_, May 18 2015