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

1,1

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

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.

Formula

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

Example

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

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 *)

Prog

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

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).

Sequence in context: A157822 A034616 A084551 * A276820 A273815 A258885

Adjacent sequences: A206042 A206043 A206044 * A206046 A206047 A206048

Keyword

nonn

Author

Sameen Ahmed Khan, Feb 03 2012

Extensions

New name from Charles R Greathouse IV, May 18 2015

Status

approved