A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

2, 2, 3, 251, 9843019, 121174811

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n-1). a(6) was found by Lander and Parkin. The next term requires a spacing >= 210. The expected size is a(7) > 10^21 (see link). - Hugo Pfoertner, Jun 25 2004

From Daniel Forgues, Jan 17 2011: (Start)

It is conjectured that there are arithmetic progressions of n consecutive primes for any n.

Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)

a(7) <= 71137654873189893604531, found by P. Zimmermann, cf. J. K. Andersen link. - Bert Dobbelaere, Jul 27 2022

Examples

			First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:
a(1) = 2: (2) (degenerate arithmetic progression);
a(2) = 2: (2, 3) (degenerate arithmetic progression);
a(3) = 3: (3, 5, 7);
a(4) = 251: (251, 257, 263, 269);
a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);
a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Jens Kruse Andersen, The smallest known CPAP-k.
• Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
• Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
• H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.
• Daniel Forgues, Wiki about consecutive primes in arithmetic progression.
• L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp., Vol. 21, No. 99 (1967) p 489.
• Manfred Toplic, The nine and ten primes project, 2004.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A005115, A006562, A093364, A126989.

a(5) corresponds to A052243(20) followed by A052243(21) 9843049.

Cf. A089180: indices primes a(n).

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4), A033451: start of CPAP-4 with common difference 6, A052239: start of first CPAP-4 with common difference 6n.

Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60.

Cf. A058362: start of 6 consecutive primes in arithmetic progression.

Programs

Mathematica

Join[{2},Table[SelectFirst[Partition[Prime[Range[691*10^4]],n,1], Length[ Union[ Differences[ #]]] == 1&][[1]],{n,2,6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)

Formula

a(n) = A000040(A089180(n)), or A089180(n) = A000720(a(n)). - M. F. Hasler, Oct 27 2018

Extensions

Edited by Daniel Forgues, Jan 17 2011