This site is supported by donations to The OEIS Foundation. 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) 53
 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 From Pierre CAMI, Sep 08 2017: (Start) a(n) = prime(i), lim_{N->inf} (Sum_{n=1..N} i) / (Sum_{n=1..N} n) = 3/2. i/n is always < 6. Lim_{N->inf} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = Pi/2. a(n) / prime(n) is always < 8. (End) REFERENCES Martin Gardner, The Last Recreations (1997), pp. 194-95. R. K. Guy, Unsolved Problems in Number Theory, Section A2 Richards, Stephen P., A Number For Your Thoughts, 1982, p. 200. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Pierre CAMI, Table of n, a(n) for n = 1..3000 (first 2000 terms from T. D. Noe) C. Banderier, Conjecture checked for n < 1000 [It has been reported that the data given here contains several errors] S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210. R. K. Guy, Letter to N. J. A. Sloane, 1987 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy] 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 [math.HO], 2012. Eric Weisstein's World of Mathematics, Fortunate Prime R. G. Wilson, V, Letter to N. J. A. Sloane with attachment, Jan 1992 FORMULA If x(n) = 1 + Product_{i=1..n} prime(i), 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. a(n) = A035345(n) - A002110(n). - Jonathan Sondow, Dec 02 2015 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. MAPLE Primorial:= 2: p:= 2: A:= 3: for n from 2 to 100 do   p:= nextprime(p);   Primorial:= Primorial * p;   A[n]:= nextprime(Primorial+p+1)-Primorial; od: seq(A[n], n=1..100); # Robert Israel, Dec 02 2015 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]]]+1) (* Harvey P. Dale, Dec 15 2013 *) PROG (PARI) a(n)=my(P=prod(k=1, n, prime(k))); nextprime(P+2)-P \\ Charles R Greathouse IV, Jul 15 2011; corrected by Jean-Marc Rebert, Jul 28 2015 (Haskell) a005235 n = head [m | m <- [3, 5 ..], a010051'' (a002110 n + m) == 1] -- Reinhard Zumkeller, Apr 02 2014 (Python) from operator import mul def x(n): return 1 + reduce(mul, [prime(i) for i in xrange(1, n + 1)]) print [nextprime(x(n)) - x(n) + 1 for n in xrange(1, 101)] # Indranil Ghosh, Mar 23 2017 CROSSREFS Cf. A046066, A002110, A006862, A035345, A035346, A055211, A129912, A010051, A005408, A038771, A038711. Sequence in context: A051507 A173145 A060274 * A107664 A321671 A085013 Adjacent sequences:  A005232 A005233 A005234 * A005236 A005237 A005238 KEYWORD nonn,nice AUTHOR 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 | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 23 14:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)