A083770 -id:A083770 - cp's OEIS frontend

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

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

Offset: 1

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

nonn

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

			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

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

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

  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)];

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 *)

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

cp's OEIS Frontend

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

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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