

A005235


Fortunate numbers: least m>1 such that m+prime(n)# is prime, where p# denotes the product of all primes <= p.
(Formerly M2418)


38



3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

R. F. Fortune conjectured that a(n) is always prime.
The first 500 terms are primes  Robert G. Wilson v; (the first 2000 terms are prime, Joerg Arndt, Apr 15 2013).
The strong form of CramÃ©r's conjecture implies that a(n) is a prime for n > 1618, as previously noted by Golomb.  Charles R Greathouse IV, Jul 05 2011
a(n) is the smallest m such that m > 1 and A002110(n)+m is prime. For every n, a(n) must be greater than prime(n+1)1.  Farideh Firoozbakht, Aug 20 2003
If a(n) < prime(n+1)^2 then a(n) is prime. According to Cramer's conjecture a(n) = O(prime(n)^2).  Thomas Ordowski, Apr 09 2013
Let b(n) = smallest composite number k such that k + prime(n)# is prime. It appears that the following inequality is true: b(n) > 2*a(n) for all n.  Thomas Ordowski, Apr 14 2013


REFERENCES

Martin Gardner, The Last Recreations (1997), pp. 19495.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209210.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly, 95 (1988), 697712.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..2000 [a(n)+prime(n)# is a probable prime]
C. Banderier, Conjecture checked for n < 1000 [It has been reported that the data given here contains several errors]
Bill McEachen, McEachen Conjecture
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, Arxiv preprint arXiv:1202.3670, 2012  From N. J. A. Sloane, Jun 13 2012
Eric Weisstein's World of Mathematics, Fortunate Prime


FORMULA

If x(n) = 1 + product(prime(i), i=1..n), q(n) = least prime > x(n), then a(n) = q(n)x(n)+1.
a(n) = 1 + the difference between the nth primorial plus one and the next prime.


EXAMPLE

a(4) = 13 because P_4# = 2*3*5*7 = 210, plus one is 211, the next prime is 223 and the difference between 210 and 223 is 13.


MATHEMATICA

NPrime[n_Integer] := Module[{k}, k = n + 1; While[! PrimeQ[k], k++]; k]; Fortunate[n_Integer] := Module[{p, q}, p = Product[Prime[i], {i, 1, n}] + 1; q = NPrime[p]; q  p + 1]; Table[Fortunate[n], {n, 60}]
r[n_] := (For[m = (Prime[n + 1] + 1)/2, ! PrimeQ[Product[Prime[k], {k, n}] + 2 m  1], m++]; 2 m  1); Table[r[n], {n, 60}]
FN[n_] := Times @@ Prime[Range[n]]; Table[NextPrime[FN[k] + 1]  FN[k], {k, 60}] (* Jayanta Basu, Apr 24 2013 *)
NextPrime[#]#+1&/@(Rest[FoldList[Times, 1, Prime[Range[60]]]]+1) (* Harvey P. Dale, Dec 15 2013 *)


PROG

(PARI) a(n)=my(P=prod(k=1, n, prime(k))); nextprime(P+1)P \\ Charles R Greathouse IV, Jul 15 2011
(Haskell)
a005235 n = head [m  m < [3, 5 ..], a010051'' (a002110 n + m) == 1]
 Reinhard Zumkeller, Apr 02 2014


CROSSREFS

Cf. A046066, A002110, A006862, A035345, A035346, A055211, A129912.
Cf. A010051, A005408.
Sequence in context: A051507 A173145 A060274 * A107664 A085013 A164939
Adjacent sequences: A005232 A005233 A005234 * A005236 A005237 A005238


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

More terms from Jud McCranie.
Guy lists 100 terms, as computed by Stan Wagon.


STATUS

approved



