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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

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.

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

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.

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

Sequence in context: A177764 A027498 A094877 * A088251 A229627 A140839

Adjacent sequences:  A006557 A006558 A006559 * A006561 A006562 A006563

KEYWORD

nonn,hard,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Daniel Forgues, Jan 17 2011

STATUS

approved

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Last modified October 20 23:11 EDT 2017. Contains 293679 sequences.