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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. 5
2, 6, 8, 12, 16, 10, 4, 30, 26, 22, 24, 14, 50, 42, 18, 64, 46, 60, 32, 36, 20, 34, 28, 108, 48, 44, 68, 282, 90, 54, 76, 62, 180, 66, 132, 86, 74, 38, 58, 106, 120, 52, 244, 94, 100, 82, 138, 156, 98, 72, 172, 150, 248, 154, 166, 114, 162, 126, 124, 208, 222, 324, 212 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Is this a permutation of the even numbers?

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

LINKS

Robert Israel, Table of n, a(n) for n = 1..200

EXAMPLE

2+1=3, 2*6+1=13, 2*6*8+1=97, 2*6*8*12+1=1153, etc. are primes.

After 200 terms the prime is

224198929826405912196464851358435330956778558123234657623126\

069546460095464785674042966210907411841359152393200850271694\

899718487202330385432243578646330245831108247815285116235792\

875886417750289946171599027675234787802312202111702704952223\

563058999855839876391430601719636148884060097930252529666254\

756431522481046758186320659298713737639441014068272279177710\

551232067814381240340990584869121776471244800000000000000000\

00000000000000000000000000000 (449 digits). - Robert Israel, Dec 21 2012

MAPLE

  N := 200: # number of terms desired

P := 2:

a[1] := 2:

C := {seq(2*j, j = 2 .. 10)}:

Cmax := 20:

for n from 2 to N do

   for t in C do

      if isprime(t*P+1) then

        a[n]:= t;

        P:= t*P;

        C:= C minus {t};

        break;

      end if;

   end do;

   while not assigned(a[n]) do

     t0:= Cmax+2;

     Cmax:= 2*Cmax;

     C:= C union {seq(j, j=t0 .. Cmax, 2)};

     for t from t0 to Cmax by 2 do

       if isprime(t*P+1) then

         a[n]:= t;

         P:= t*P;

         C:= C minus {t};

         break;

       end if

     end do;

   end do;

end do;

[seq(a[n], n=1..N)];

MATHEMATICA

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 *)

CROSSREFS

Cf. A036013, A046966, A046972, A051957, A073673, A073674, A083769, A083770, A083771, A084401, A084402, A084724, A087338.

Sequence in context: A064796 A304483 A268177 * A057656 A247066 A084724

Adjacent sequences:  A083766 A083767 A083768 * A083770 A083771 A083772

KEYWORD

nonn

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

EXTENSIONS

More terms from David Wasserman, Nov 23 2004

Edited by N. J. A. Sloane, Dec 20 2012

Comment edited, Maple code and additional terms by Robert Israel, Dec 20 2012

STATUS

approved

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Last modified May 25 21:17 EDT 2022. Contains 354071 sequences. (Running on oeis4.)