OFFSET

1,1

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

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.

FORMULA

EXAMPLE

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);

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 *)

CROSSREFS

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.

KEYWORD

nonn,hard,more,nice

AUTHOR

EXTENSIONS

Edited by Daniel Forgues, Jan 17 2011

STATUS

approved