A062894 -id:A062894 - 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

A215464 Smallest prime not generated by Buss's function (A067836) through B(n).

Original entry on oeis.org

3, 5, 7, 11, 11, 17, 19, 23, 29, 29, 29, 31, 41, 41, 41, 41, 41, 41, 47, 59, 59, 59, 59, 59, 59, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 89, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97

Offset: 1

Robert Price, Mar 09 2013

nonn

			a(5)=11 since B(5)=13, but the prime 11 has not yet appeared in the Buss's function sequence.

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A067836, A062894, A067949, A215463.

A364824 Index of prime(n) in A067836, or -1 if prime(n) does not occur in it.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 9, 12, 13, 10, 19, 14, 20, 16, 26, 42, 18, 25, 11, 15, 17, 43, 118, 33, 24, 31, 29, 212, 40, 68, 30, 98, 22, 45, 34, 109, 28, 39, 21, 23, 46, 143, 27, 35, 37, 55, 47, 123, 58, 90, 132, 32, 139, 91, 41, 44, 38, 52, 36, 77, 54, 48, 53, 83, 51

Offset: 1

Bert Dobbelaere, Aug 09 2023

nonn

All terms in A067836 are distinct, making this sequence defined.
If the conjecture holds that A067836 contains only primes, then a(A062894(n)) = n.
If all primes eventually occur in A067836, then all terms in this sequence are positive and A062894(a(n)) = n.

Cf. A062894, A067836.

from itertools import count
from sympy import prime, nextprime
def A364824(n):
    a, f, p = 2, 1, prime(n)
    for i in count(1):
        if a == p:
            return i
        a=nextprime((f:=f*a)+1)-f # Chai Wah Wu, Sep 09 2023

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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A215464 Smallest prime not generated by Buss's function (A067836) through B(n).

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A364824 Index of prime(n) in A067836, or -1 if prime(n) does not occur in it.

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Python

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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Author

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Comments

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Crossrefs

Programs

Mathematica

MuPAD

PARI

Python

Extensions

A215464 Smallest prime not generated by Buss's function (A067836) through B(n).

Views

Author

Keywords

Examples

Links

Crossrefs

A364824 Index of prime(n) in A067836, or -1 if prime(n) does not occur in it.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

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.