login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 Cramér'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. 194-95.

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

R. K. Guy, The strong law of small numbers, Amer. Math. Monthly, 95 (1988), 697-712.

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 BC--2012) 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 n-th 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified July 29 21:54 EDT 2014. Contains 245046 sequences.