A206045 Numbers d such that 11 + j*d is prime for j = 0 to 10.

1536160080, 4911773580, 25104552900, 77375139660, 83516678490, 100070721660, 150365447400, 300035001630, 318652145070, 369822103350, 377344636200, 511688932650, 580028072610, 638663371710, 701534299830, 745828915650, 776625236100, 883476548850, 925639075620, 956863233690

Offset: 1

Author

Sameen Ahmed Khan, Feb 03 2012

Keywords

• nonn

Comments

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.

The computations were done without any assumptions on the form of d. 21st term is greater than 10^12.

All terms are multiples of 210=2*3*5*7. - Zak Seidov, May 16 2015

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

Examples

			d = 4911773580 then {11, 4911773591, 9823547171, 14735320751, 19647094331, 24558867911, 29470641491, 34382415071, 39294188651, 44205962231, 49117735811} which is 11 primes in arithmetic progression.
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 139.

Links

• Zak Seidov, Table of n, a(n) for n = 1..623.
• 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.
• Wikipedia, Primes in arithmetic progression.
• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

Cf. A040976, A206038, A206040, A206042, A206043, A206044.

Cf. A123556, A173919.

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).

Programs

Mathematica

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 *)
Select[Range[210,10^12,210],AllTrue[Range[0,10]#+11,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2016 *)

PARI

is(n)=for(j=1,10, if(!isprime(j*n+11), return(0))); 1 \\ Charles R Greathouse IV, May 18 2015

Formula

m is a term iff A123556(m) = 11. - Bernard Schott, Mar 08 2023

Extensions

New name from Charles R Greathouse IV, May 18 2015