A067836 Let a(1)=2, f(n)=a(1)*a(2)*...*a(n-1) for n>=1 and a(n)=nextprime(f(n)+1)-f(n) for n>=2, where nextprime(x) is the smallest prime > x.

2, 3, 5, 7, 13, 11, 17, 19, 23, 37, 73, 29, 31, 43, 79, 53, 83, 67, 41, 47, 179, 149, 181, 103, 71, 59, 197, 167, 109, 137, 107, 251, 101, 157, 199, 283, 211, 277, 173, 127, 269, 61, 89, 271, 151, 191, 227, 311, 409, 577, 331, 281, 313, 307, 223, 491, 439, 233, 367

Offset

1,1

Comments

The terms are easily seen to be distinct. It is conjectured that every element is prime. Do all primes occur in the sequence?

All elements are prime and distinct through n=1000. - Robert Price, Mar 09 2013

All elements are prime and distinct through n=3724. - Dana Jacobsen, Feb 15 2015

With a(0) = 1, a(n) is the next smallest number not in the sequence such that a(n) + Product_{i=1..n-1} a(i) is prime. - Derek Orr, Jun 16 2015

Links

Robert Price and Dana Jacobsen, Table of n, a(n) for n = 1..3724 (first 1000 terms from Robert Price)

Frank Buss, Prime Puzzles - Frank Buss's Conjecture

Frank Buss, The B(n) function

Mathematica

<<NumberTheory`PrimeQ` (* Load ProvablePrimeQ function, needed below. *)
f[1]=1; f[n_] := f[n]=f[n-1]a[n-1]; a[n_] := a[n]=Module[{i}, For[i=2, True, i++, If[ProvablePrimeQ[f[n]+i], Return[i]]]]
Join[{a = 2}, f = 1; Table[f = f*a; a = NextPrime[f + 1] - f; a, {n, 2, 59}]] (* Jayanta Basu, Aug 10 2013 *)

Prog

(MuPAD) f := 1:for n from 1 to 50 do a := nextprime(f+2)-f:f := f*a:print(a) end_for
(PARI) v=[2]; n=2; while(n<500, s=n+prod(i=1, #v, v[i]); if(isprime(s)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=1); n++); v \\ Derek Orr, Jun 16 2015
(Python)
from sympy import nextprime
def A067836_gen(): # generator of terms
    a, f = 2, 1
    yield 2
    while True:
        yield (a:=nextprime((f:=f*a)+1)-f)
A067836_list = list(islice(A067836_gen(), 30)) # Chai Wah Wu, Sep 09 2023

Crossrefs

Cf. A062894 has the indices of the primes in this sequence. A071290 has the sequence of f's. Also see A067362, A068192.

Sequence in context: A126056 A126055 A126054 * A332806 A108546 A065107

Adjacent sequences: A067833 A067834 A067835 * A067837 A067838 A067839

Keyword

nonn

Author

Frank Buss (fb(AT)frank-buss.de), Feb 09 2002

Extensions

Edited by Dean Hickerson, Mar 02 2002

Edited by Dean Hickerson and David W. Wilson, Jun 10 2002

Status

approved