A206037 - OEIS

%I #50 Jan 29 2023 19:00:08

%S 2,4,8,10,14,20,28,34,38,40,50,64,68,80,94,98,104,110,124,134,154,164,

%T 178,188,190,208,220,230,238,248,260,280,308,314,328,344,370,418,428,

%U 430,440,454,458,484,518,544,560,574,584,610,614,628,638,640,644,650

%N Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.

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

%C Numbers k such that k+3 and 2k+3 are both primes.

%C Equivalently, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly 3 elements (see example). These 3 elements are not necessarily consecutive primes. In fact, for each term d, there exists only one such AP of primes, and this one starts always with A342309(d) = 3, so this AP is (3, 3+d, 3+2d). - _Bernard Schott_, Jan 15 2023

%H Sameen Ahmed Khan, <a href="/A206037/b206037.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 A. 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 a(n) = 2 * A115334(n). - _Wesley Ivan Hurt_, Feb 06 2014

%F m is a term iff A123556(m) = 3. - _Bernard Schott_, Jan 15 2023

%e d = 8 then {3, 3 + 1*8, 3 + 2*8} = {3, 11, 19}, which is 3 primes in arithmetic progression.

%p filter := d -> isprime(3+d) and isprime(3+2*d) : select(filter, [$(1 .. 650)]); # _Bernard Schott_, Jan 16 2023

%t t={}; Do[If[PrimeQ[{3, 3 + d, 3 + 2*d}] == {True, True, True}, AppendTo[t, d]], {d, 1000}]; t

%t Select[Range[2,700,2],And@@PrimeQ[{3+#,3+2#}]&] (* _Harvey P. Dale_, Sep 25 2013 *)

%o (TI-Basic) Clrio:Input "n",n:Lbl colorin:If isPrime(n+3) and isPrime(2*n+3) Then:Disp n:Pause:Endif:n+1(sto)n:Goto colorin:EndPrgm " _César Aguilera_, Dec 27 2015

%o (PARI) for(n=1, 1e3, if(isprime(n + 3) && isprime(2*n + 3), print1(n, ", "))); \\ _Altug Alkan_, Dec 27 2015

%o (Magma) [n: n in [1..700] | IsPrime(3+n) and IsPrime(3+2*n)]; // _Vincenzo Librandi_, Dec 28 2015

%Y Cf. A040976, A123556, A206038, A206039, A206040, A206041, A206042, A206043, A206044, A206045, A342309.

%Y Largest AP of prime numbers with k elements: A007921 (k=1), A359408 (k=2), this sequence (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7).

%K nonn,easy

%O 1,1

%A _Sameen Ahmed Khan_, Feb 03 2012