A206037 Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.

2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 64, 68, 80, 94, 98, 104, 110, 124, 134, 154, 164, 178, 188, 190, 208, 220, 230, 238, 248, 260, 280, 308, 314, 328, 344, 370, 418, 428, 430, 440, 454, 458, 484, 518, 544, 560, 574, 584, 610, 614, 628, 638, 640, 644, 650

Offset: 1

Author

Sameen Ahmed Khan, Feb 03 2012

Keywords

• nonn
• easy

Comments

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

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

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

Examples

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

Links

• Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000.
• Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
• Sameen A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
• Wikipedia, Primes in arithmetic progression.
• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

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

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).

Programs

Magma

[n: n in [1..700] | IsPrime(3+n) and IsPrime(3+2*n)]; // Vincenzo Librandi, Dec 28 2015

Maple

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

Mathematica

t={}; Do[If[PrimeQ[{3, 3 + d, 3 + 2*d}] == {True, True, True}, AppendTo[t, d]], {d, 1000}]; t
Select[Range[2,700,2],And@@PrimeQ[{3+#,3+2#}]&] (* Harvey P. Dale, Sep 25 2013 *)

PARI

for(n=1, 1e3, if(isprime(n + 3) && isprime(2*n + 3), print1(n, ", "))); \\ Altug Alkan, Dec 27 2015

Formula

a(n) = 2 * A115334(n). - Wesley Ivan Hurt, Feb 06 2014

m is a term iff A123556(m) = 3. - Bernard Schott, Jan 15 2023