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

6, 12, 42, 48, 96, 126, 252, 426, 474, 594, 636, 804, 1218, 1314, 1428, 1566, 1728, 1896, 2106, 2574, 2694, 2898, 3162, 3366, 4332, 4368, 4716, 4914, 4926, 4962, 5472, 5586, 5796, 5838, 6048, 7446, 7572, 7818, 8034, 8958, 9168, 9204, 9714

Offset: 1

Author

Sameen Ahmed Khan, Feb 03 2012

Keywords

• nonn

Comments

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

All terms are multiples of 6. - Zak Seidov, Jan 07 2014

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

Examples

			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.
		

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 Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012
• Wikipedia, Primes in arithmetic progression.
• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

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

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).

Programs

Maple

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

Mathematica

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
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 *)

Formula

m is a term iff A123556(m) = 3. - Bernard Schott, Jan 25 2023