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A083769 a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)*a(2)*...*a(n) + 1 is prime.

%I A083769 #40 Aug 23 2014 13:06:34
%S A083769 2,6,8,12,16,10,4,30,26,22,24,14,50,42,18,64,46,60,32,36,20,34,28,108,
%T A083769 48,44,68,282,90,54,76,62,180,66,132,86,74,38,58,106,120,52,244,94,
%U A083769 100,82,138,156,98,72,172,150,248,154,166,114,162,126,124,208,222,324,212
%N A083769 a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)*a(2)*...*a(n) + 1 is prime.
%C A083769 Is this a permutation of the even numbers?
%C A083769 For any even positive integers a_1, a_2, ..., a_n, there are infinitely many even positive integers t such that a_1 a_2 ... a_n t + 1 is prime: this follows from Dirichlet's theorem on primes in arithmetic progressions.  As far as I know there is no guarantee that the sequence defined here leads to a permutation of the even numbers, i.e. there might be some even integer that never appears in the sequence. However, if the partial products a_1 ... a_n grow like 2^n n!, heuristically the probability of a_1 ... a_n t + 1 being prime is on the order of 1/log(a_1 ... a_n) ~ 1/(n log n), and since sum_n 1/(n log n) diverges we might expect that there should be infinitely many n for which some a_1 ... a_n t + 1 is prime, and thus every even integer should occur. - _Robert Israel_, Dec 20 2012
%H A083769 Robert Israel, <a href="/A083769/b083769.txt">Table of n, a(n) for n = 1..200</a>
%e A083769 2+1=3, 2*6+1=13, 2*6*8+1=97, 2*6*8*12+1=1153, etc. are primes.
%e A083769 After 200 terms the prime is
%e A083769 224198929826405912196464851358435330956778558123234657623126\
%e A083769 069546460095464785674042966210907411841359152393200850271694\
%e A083769 899718487202330385432243578646330245831108247815285116235792\
%e A083769 875886417750289946171599027675234787802312202111702704952223\
%e A083769 563058999855839876391430601719636148884060097930252529666254\
%e A083769 756431522481046758186320659298713737639441014068272279177710\
%e A083769 551232067814381240340990584869121776471244800000000000000000\
%e A083769 00000000000000000000000000000 (449 digits). - _Robert Israel_, Dec 21 2012
%p A083769   N := 200: # number of terms desired
%p A083769 P := 2:
%p A083769 a[1] := 2:
%p A083769 C := {seq(2*j, j = 2 .. 10)}:
%p A083769 Cmax := 20:
%p A083769 for n from 2 to N do
%p A083769    for t in C do
%p A083769       if isprime(t*P+1) then
%p A083769         a[n]:= t;
%p A083769         P:= t*P;
%p A083769         C:= C minus {t};
%p A083769         break;
%p A083769       end if;
%p A083769    end do;
%p A083769    while not assigned(a[n]) do
%p A083769      t0:= Cmax+2;
%p A083769      Cmax:= 2*Cmax;
%p A083769      C:= C union {seq(j, j=t0 .. Cmax, 2)};
%p A083769      for t from t0 to Cmax by 2 do
%p A083769        if isprime(t*P+1) then
%p A083769          a[n]:= t;
%p A083769          P:= t*P;
%p A083769          C:= C minus {t};
%p A083769          break;
%p A083769        end if
%p A083769      end do;
%p A083769    end do;
%p A083769 end do;
%p A083769 [seq(a[n],n=1..N)];
%t A083769 f[s_List] := Block[{k = 2, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k += 2]; Append[s, k]]; Nest[f, {2}, 62] (* _Robert G. Wilson v_, Dec 24 2012 *)
%Y A083769 Cf. A036013, A046966, A046972, A051957, A073673, A073674, A083769, A083770, A083771, A084401, A084402, A084724, A087338.
%K A083769 nonn
%O A083769 1,1
%A A083769 _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003
%E A083769 More terms from _David Wasserman_, Nov 23 2004
%E A083769 Edited by _N. J. A. Sloane_, Dec 20 2012
%E A083769 Comment edited, Maple code and additional terms by _Robert Israel_, Dec 20 2012