|
|
A005234
|
|
Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime.
(Formerly M0669)
|
|
23
|
|
|
2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Conjecture: if p# + 1 is a prime number, then the next prime is less than p# + exp(1)*p. - Arkadiusz Wesolowski, Feb 20 2013
Conjecture: if p# + 1 is a prime, then the next prime is less than p# + p^2. - Thomas Ordowski, Apr 07 2013
|
|
REFERENCES
|
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 109, 1983.
Paulo Ribenboim, The New Book of Prime Number Records, p. 13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 112.
|
|
LINKS
|
|
|
FORMULA
|
|
|
MAPLE
|
N:= 5000: # to get all terms <= N
Primes:= select(isprime, [$2..N]):
P:= 1: count:= 0:
for n from 1 to nops(Primes) do
P:= P*Primes[n];
if isprime(P+1) then
count:= count+1; A[count]:= Primes[n]
fi
od:
|
|
MATHEMATICA
|
(* This program is not convenient for large values of p *) p = pp = 1; Reap[While[p < 5000, p = NextPrime[p]; pp = pp*p; If[PrimeQ[1 + pp], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 31 2012 *)
With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] + 1, _?PrimeQ]]]]] (* Eric W. Weisstein, Nov 03 2015 *)
|
|
PROG
|
(PARI) is(n)=isprime(n) && ispseudoprime(prod(i=1, primepi(n), prime(i))+1) \\ Charles R Greathouse IV, Feb 20 2013
(PARI) is(n)=isprime(n) && ispseudoprime(factorback(primes([2, n]))+1) \\ M. F. Hasler, May 31 2018
(Magma) [p:p in PrimesUpTo(3000)|IsPrime(&*PrimesUpTo(p)+1)]; // Marius A. Burtea, Mar 25 2019
|
|
CROSSREFS
|
Cf. A014545 (Primorial plus 1 prime indices: n such that 1 + (Product of first n primes) is prime).
Cf. A018239 (Primorial plus 1 primes).
|
|
KEYWORD
|
nonn,hard,more,nice,changed
|
|
AUTHOR
|
|
|
EXTENSIONS
|
42209 sent in by Chris Nash (chrisnash(AT)cwix.com).
145823 discovered and sent in by Arlin Anderson (starship1(AT)gmail.com) and Don Robinson (donald.robinson(AT)itt.com), Jun 01 2000
|
|
STATUS
|
approved
|
|
|
|