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%I #19 Mar 01 2022 09:28:41
%S 1,1,2,330,10830,25410,512820,512820,12960606120,434491727670,
%T 1893245380950,71023095613470,878232256181280,11429352906540438870
%N a(n) is the smallest number k such that r*k + 1 is prime for all r = 1 to n.
%C Conjectures: (1) Sequence is infinite. (2) For every n there are infinitely many arithmetic progressions with n successive primes.
%C Both conjectures follow from Dickson's conjecture. [_Charles R Greathouse IV_, Mar 14 2011]
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_379.htm">Puzzle 379. SG primes and its prime average</a>, The Prime Puzzles and Problems Connection.
%e a(11) = 1893245380950 because all eleven numbers 1*1893245380950 + 1, 2*1893245380950 + 1, 3*1893245380950 + 1, ..., 10*1893245380950 + 1 & 11*1893245380950 + 1 are prime and 1893245380950 is the smallest number with such property.
%t a[n_] := Block[{k = If[n < 4, 1, 6], s}, s = k; While[! AllTrue[k Range[n] + 1, PrimeQ], k += s]; k]; Array[a, 8] (* _Giovanni Resta_, Mar 31 2017 *)
%Y Cf. A088251, A088252, A088651, A125838, A125839.
%K nonn,more
%O 1,3
%A _Amarnath Murthy_, Sep 26 2003
%E Edited by _Don Reble_, Sep 29 2003
%E Entry revised by _N. J. A. Sloane_, Jan 05 2007
%E a(14) from _Giovanni Resta_, Mar 31 2017
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