

A006560


Smallest starting prime for n consecutive primes in arithmetic progression.
(Formerly M0927)


9




OFFSET

1,1


COMMENTS

The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n1). 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) <= 382003672700092872707633, found by P. Zimmermann, cf. J. K. Andersen link.  M. F. Hasler, Oct 26 2018


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 CPAPk.
Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 17431749. MR 98a:11122.
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 13231328.
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


FORMULA

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


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. 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 (CPAP4), A033451: start of CPAP4 with common difference 6, A052239: start of first CPAP4 with common difference 6n.
Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP5 with common difference 60.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.
Sequence in context: A177764 A027498 A094877 * A088251 A229627 A140839
Adjacent sequences: A006557 A006558 A006559 * A006561 A006562 A006563


KEYWORD

nonn,hard,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Edited by Daniel Forgues, Jan 17 2011


STATUS

approved



