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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.
11
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
A generalization of Fortunate's conjecture, cf. A005235. - M. F. Hasler, Nov 04 2024
LINKS
Robert Price and Dana Jacobsen, Table of n, a(n) for n = 1..3724 (first 1000 terms from Robert Price)
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.
Cf. A005235 (Fortunate numbers).
Sequence in context: A126056 A126055 A126054 * A332806 A108546 A065107
KEYWORD
nonn,changed
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