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%I #29 Nov 11 2019 03:04:25
%S 2,5,7,13,37,73,7937,7703,272809,640943,5378959,116137159,3708797237,
%T 114649314209,158317270283
%N First term of first sequence of n primes in arithmetic progression with a common difference equal to the product of first n primes.
%C a(14) > 2^32 and a(15) > 2^32. - _Jud McCranie_
%H R. Chapman, <a href="https://empslocal.ex.ac.uk/people/staff/rjchapma/etc/dirichlet.pdf">Dirichlet's theorem: a real variable approach</a>, 2008.
%H B. Green & T. Tao, <a href="https://arxiv.org/abs/math/0404188">The primes contain arbitrarily long arithmetic progressions</a>, arXiv:math/0404188 [math.NT], 2004-2007.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%e For n=3, product of first 3 primes is 30. The first arithmetic progression of 3 primes with difference 30 starts at 7 (7, 37, 67), so a(3)=7.
%t (* This program is not convenient beyond 10 terms *) r[p1_, n_] := Reduce[p[1] = p1; Equal @@ Append[Table[p[k + 1] - p[k], {k, 1, n - 1}], Product[Prime[k], {k, 1, n}]], p[2], Primes]; a[n_] := a[n] = Catch[For[k = 1, k <= 10^5, k++, If[r[p1 = Prime[k], n] =!= False, Throw[p1]]]]; Table[Print[a[n]]; a[n], {n, 1, 10}] (* _Jean-François Alcover_, Dec 27 2012 *)
%K hard,nonn,nice
%O 1,1
%A _G. L. Honaker, Jr._, Feb 18 2000
%E Last 3 terms from _Jud McCranie_, Feb 28 2000
%E a(14)-a(15) from _Donovan Johnson_, Oct 20 2009
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