A071290 -id:A071290 - cp's OEIS frontend

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

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

nonn

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
A generalization of Fortunate's conjecture, cf. A005235. - M. F. Hasler, Nov 04 2024
Conjecture: there are infinitely many values of this sequence such that a(n) < n. - Davide Rotondo, Feb 28 2025

Robert Price and Dana Jacobsen, Table of n, a(n) for n = 1..3724 (first 1000 terms from Robert Price)
Carlos Rivera, Conjecture 28. Frank Buss's Conjecture, The Prime Puzzles & Problems Connection.
Carlos Rivera, Conjecture 29. The Frank Buss's B(n) function, The Prime Puzzles & Problems Connection.

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

Cf. A005235 (Fortunate numbers).

Mathematica
```
<Jayanta Basu, Aug 10 2013 *)
```

f := 1:for n from 1 to 50 do a := nextprime(f+2)-f:f := f*a:print(a) end_for

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

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

Edited by Dean Hickerson, Mar 02 2002

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

cp's OEIS Frontend

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.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

MuPAD

PARI

Python

Extensions

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.