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%I A005235 M2418
%S A005235 3,5,7,13,23,17,19,23,37,61,67,61,71,47,107,59,61,109,89,103,79,151,197,
%T A005235 101,103,233,223,127,223,191,163,229,643,239,157,167,439,239,199,191,199,
%U A005235 383,233,751,313,773,607,313,383,293,443,331,283,277,271,401,307,331
%N A005235 Fortunate numbers: least m>1 such that m+prime(n)# is prime, where p# 
               denotes the product of the primes <= p.
%C A005235 R. F. Fortune conjectured that a(n) is always prime.
%C A005235 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 
               (f.firoozbakht(AT)sci.ui.ac.ir), Aug 20 2003
%D A005235 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005235 Martin Gardner, The Last Recreations (1997), pp. 194-95.
%D A005235 S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 
               209-210.
%D A005235 R. K. Guy, Unsolved Problems in Number Theory, Section A2.
%D A005235 R. K. Guy, The strong law of small numbers, Amer. Math. Monthly, 95 (1988), 
               697-712.
%H A005235 T. D. Noe, <a href="b005235.txt">Table of n, a(n) for n=1..2000</a> (a(n)+prime(n)# 
               is a probable prime)
%H A005235 C. Banderier, <a href="http://algo.inria.fr/banderier/Computations/prime_factorial.html">
               Conjecture checked for n<1000</a> [It has been reported that that 
               the data given here contains several errors]
%H A005235 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FortunatePrime.html">Link to a section of The World of Mathematics.</
               a>
%H A005235 Author?, <a href="http://garden.irmacs.sfu.ca/?q=op/maceachen_conjecture">
               McEachen Conjecture</a>
%F A005235 If x(n) = 1 + product(prime(i), i=1..n), q(n) = least prime > x(n), then 
               a(n) = q(n)-x(n)+1.
%F A005235 a(n) = 1 + the difference between the n-th Primorial plus one and the 
               next prime.
%e A005235 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.
%t A005235 NextPrime[ 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 = NextPrime[ p ]; q - p + 1 ]; Table[ 
               Fortunate[ n ], {n, 1, 60} ]
%t A005235 r[n_] := (For[m=(Prime[n+1]+1)/2, !PrimeQ[Product[Prime[k], {k, n}]+2m-1], 
               m++ ]; 2m-1); Table[a[n], {n, 60}]
%Y A005235 Cf. A046066, A002110, A006862, A035345, A035346, A055211.
%Y A005235 Cf. A129912.
%Y A005235 Sequence in context: A154700 A051507 A060274 this_sequence A107664 A085013 
               A164939
%Y A005235 Adjacent sequences: A005232 A005233 A005234 this_sequence A005236 A005237 
               A005238
%K A005235 nonn,nice
%O A005235 1,1
%A A005235 N. J. A. Sloane (njas(AT)research.att.com).
%E A005235 More terms from Jud McCranie, j.mccranie(AT)comcast.net.
%E A005235 Guy lists 100 terms, as computed by Stan Wagon.
%E A005235 The first 500 terms are primes - Robert G. Wilson v

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