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 A006560 Smallest starting prime for n consecutive primes in arithmetic progression. (Formerly M0927) 9

%I M0927

%S 2,2,3,251,9843019,121174811

%N Smallest starting prime for n consecutive primes in arithmetic progression.

%C 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

%C From _Daniel Forgues_, Jan 17 2011: (Start)

%C It is conjectured that there are arithmetic progressions of n consecutive primes for any n.

%C Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)

%C a(7) <= 382003672700092872707633, found by P. Zimmermann, cf. J. K. Andersen link. - _M. F. Hasler_, Oct 26 2018

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

%H Jens Kruse Andersen, <a href="http://primerecords.dk/cpap.htm#smallest">The smallest known CPAP-k.</a>

%H Chris K. Caldwell, <a href="http://primes.utm.edu/top20/page.php?id=13">Consecutive Primes in Arithmetic Progression</a>

%H Harvey Dubner and Harry Nelson, <a href="https://doi.org/10.1090/S0025-5718-97-00875-2">Seven consecutive primes in arithmetic progression</a>, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.

%H H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, P. Zimmermann, <a href="https://doi.org/10.1090/S0025-5718-01-01374-6">Ten consecutive primes in arithmetic progression</a>, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.

%H Daniel Forgues, <a href="http://oeis.org/wiki/Consecutive primes in arithmetic progression">Wiki about consecutive primes in arithmetic progression</a>.

%H L. J. Lander and T. R. Parkin, <a href="https://doi.org/10.1090/S0025-5718-67-99657-3">Consecutive primes in arithmetic progression</a>, Math. Comp., Vol. 21, No. 99 (1967) p 489.

%H Manfred Toplic, <a href="http://members.aon.at/toplicm/cp09.html">The nine and ten primes project</a>, 2004.

%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>

%F a(n) = A000040(A089180(n)), or A089180(n) = A000720(a(n)). - _M. F. Hasler_, Oct 27 2018

%e First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:

%e a(1) = 2: (2) (degenerate arithmetic progression);

%e a(2) = 2: (2, 3) (degenerate arithmetic progression);

%e a(3) = 3: (3, 5, 7);

%e a(4) = 251: (251, 257, 263, 269);

%e a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);

%e a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);

%Y Cf. A005115, A006562, A093364, A126989.

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

%Y Cf. A089180: indices primes a(n).

%Y 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.

%Y Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60.

%Y Cf. A058362: start of 6 consecutive primes in arithmetic progression.

%K nonn,hard,nice

%O 1,1

%A _N. J. A. Sloane_

%E Edited by _Daniel Forgues_, Jan 17 2011

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Last modified January 22 05:11 EST 2019. Contains 319353 sequences. (Running on oeis4.)