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A006560 Smallest starting prime for n consecutive primes in arithmetic progression.
(Formerly M0927)
6
2, 2, 3, 251, 9843019, 121174811 (list; graph; refs; listen; history; text; internal format)
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)

REFERENCES

L. J. Lander and T. R. Parkin, Consecutive Primes in Arithmetic Progression. Math. Comput. 21, 489, 1967.

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

LINKS

Table of n, a(n) for n=1..6.

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, P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp. 71 (239) (2002) 1323-1328.

Daniel Forgues, Wiki about consecutive primes in arithmetic progression.

Manfred Toplic, The nine and ten primes project, 2004.

Index entries for sequences related to primes in arithmetic progressions

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);

CROSSREFS

Cf. A005115, A006562, A093364, A126989.

Sequence in context: A177764 A027498 A094877 * A088251 A229627 A140839

Adjacent sequences:  A006557 A006558 A006559 * A006561 A006562 A006563

KEYWORD

nonn,hard

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Daniel Forgues, Jan 17 2011

STATUS

approved

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Last modified July 25 04:27 EDT 2017. Contains 289779 sequences.