A036012 a(n) = smallest number > 1 such that a(1)a(2)...a(n) + 1 is prime.

2, 2, 3, 3, 2, 6, 3, 2, 4, 7, 7, 3, 8, 6, 2, 3, 6, 9, 6, 14, 19, 11, 4, 4, 19, 4, 13, 3, 10, 13, 15, 4, 11, 9, 2, 5, 26, 19, 52, 21, 20, 63, 4, 19, 17, 6, 29, 19, 3, 5, 51, 11, 14, 15, 7, 12, 44, 34, 7, 21, 32, 3, 22, 10, 19, 19, 7, 20, 4, 22, 4, 17, 35, 47, 40, 14, 5, 14, 36, 39, 16

Offset: 1

Author

Russell Easterly

Keywords

• nonn
• nice
• easy

Comments

Except for the first term, same as A084401. - David Wasserman, Dec 22 2004

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..300 from T. D. Noe)

Crossrefs

Equals A084716(n+1)/A084716(n).

Cf. A036013, A084716, A084717, A084718.

Programs

Maple

n := 1: while true do j := 2: while not isprime(j*n+1) do j := j+1: od: print(j): n := n*j: od:

Mathematica

a[1] = 2; a[n_] := a[n] = Catch[For[an = 2, True, an++, If[PrimeQ[Product[a[k], {k, 1, n - 1}]*an + 1], Throw[an]]]]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Nov 27 2012 *)
nxt[{t_,n_}]:=Module[{k=2},While[!PrimeQ[t*k+1],k++];{t*k,k}]; NestList[ nxt,{2,2},80][[All,2]] (* Harvey P. Dale, Oct 03 2020 *)

Python

from gmpy2 import is_prime
from itertools import count, islice
def agen(): # generator of terms
    p = 1
    while True:
        an = next(k for k in count(2) if (t:=p*k+1) == 1 or is_prime(t))
        p *= an
        yield an
print(list(islice(agen(), 81))) # Michael S. Branicky, Jan 20 2024

Formula

Conjecture: a(n) = O(n). - Thomas Ordowski, Aug 08 2017

Extensions

More terms from Erich Friedman

More terms from Jud McCranie, Jan 26 2000

Description corrected by Len Smiley