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A206039 Values of the difference d for 5 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 4. 15

%I #40 Jan 29 2023 19:18:22

%S 6,12,42,48,96,126,252,426,474,594,636,804,1218,1314,1428,1566,1728,

%T 1896,2106,2574,2694,2898,3162,3366,4332,4368,4716,4914,4926,4962,

%U 5472,5586,5796,5838,6048,7446,7572,7818,8034,8958,9168,9204,9714

%N Values of the difference d for 5 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 4.

%C The computations were done without any assumptions on the form of d.

%C All terms are multiples of 6. - _Zak Seidov_, Jan 07 2014

%C Equivalently, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly 5 elements (see example). These 5 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) = 5, so this unique AP is (5, 5+d, 5+2d, 5+3d, 5+4d). - _Bernard Schott_, Jan 25 2023

%H Sameen Ahmed Khan, <a href="/A206039/b206039.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) = 3. - _Bernard Schott_, Jan 25 2023

%e d = 12 then {5, 5 + 1*12, 5 + 2*12, 5 + 3*12, 5 + 4*12} = {5, 17, 29, 41, 53}, which is 5 primes in arithmetic progression.

%p filter := d -> isprime(5+d) and isprime(5+2*d) and isprime(5+3*d) and isprime(5+4*d) : select(filter, [$(1 .. 10000)]); # _Bernard Schott_, Jan 25 2023

%t t={}; Do[If[PrimeQ[{5, 5 + d, 5 + 2*d, 5 + 3*d, 5 +4*d}] == {True, True, True, True, True}, AppendTo[t, d]], {d, 10000}]; t

%t Select[Range[10000],AllTrue[5+#*Range[0,4],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, May 09 2015 *)

%Y Cf. A040976, A123556, A206038, A206040, A206042, A206043, A206044, A206045, A342309.

%Y Largest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), this sequence (k=5), A359410 (k=6), A206041 (k=7), A360146 (k=10), A206045 (k=11).

%K nonn

%O 1,1

%A _Sameen Ahmed Khan_, Feb 03 2012

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)