A126989 -id:A126989 - cp's OEIS frontend

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

2, 2, 3, 251, 9843019, 121174811

Offset: 1

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

			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);

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

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

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.

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 *)

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

Edited by Daniel Forgues, Jan 17 2011

A093364 Gaps associated with the arithmetic progressions of primes in A005115.

Original entry on oeis.org

0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 13860, 13860, 60060, 420420, 4144140, 9699690, 87297210, 717777060, 4180566390, 18846497670, 26004868890

Offset: 1

Hugo Pfoertner, Apr 27 2004

See A005115 for comments, references, sources, related sequences.

Index entries for sequences related to primes in arithmetic progressions

Cf. A126989, A005115, A006560.

Entry revised by N. J. A. Sloane Jan 26 2006 based on A005115.

cp's OEIS Frontend

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

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